help-49

[cloud bronze]

So I have to worry if the equations range includes 0?

[runic hamlet]

also you can write it instead as x^3+6x^2+7x+3=(x^2+4x-1)(x+2)+5

this is also true for all x

[slate ferry]

the logic is, you cant have 0 in the denominator. so if the denominator can equal 0 for some value of x then it shall be restricted and excluded. if the denominator cannot equal to (such as dena's example) yiou dont have to worry or exclude anything

[cloud bronze]

How did u do that

[runic hamlet]

how did I do what

[cloud bronze]

Simplify the top numerator

I thought it could not be further simplified

[runic hamlet]

I did the division

[acoustic bane]

polynomial long division

[runic hamlet]

the result is x^2+4x-1, the remainder is 5

[cloud bronze]

Oh oh yeah sorry

[runic hamlet]

I just rewrote what that means

[cloud bronze]

But in that case

[runic hamlet]

into a form which doesnt use division

[cloud bronze]

That is for x is not equal to -2

[runic hamlet]

so its valid for all x

even for x=-2

[cloud bronze]

Wait

So to clarify

If u have

X^2-1

Divided by x-1

[slate ferry]

equations dont have ranges.
if you mean a function ; duivison by 0 is undefined and you cant have undefined values in the range.

if you're talking about equations ; you cannot have a solution for an equation with a 0 in the denominator.

some logic just differnet phrasing

[cloud bronze]

It is x+1 with the value 1 being restricted

[slate ferry]

oh yeah

[runic hamlet]

but the equality x^2-1=(x+1)(x-1) is always true

[slate ferry]

and ontop of that, when we turn x^2 -1 / x-1 into x+1 by cancelling the two (x-1)s we are assuming that x-1 isnt equal to 0

because we cannot cancel 0 with 0

0/0 is also a no no

thats why for equations, if we see a fraction or an even root, the first thing we do, just to clarify future solutions, is get the values x cant be.

because sometimes, you might start an equation by saying "x CANNOT be 5"

abnd then at the end you get x = 5 and x=-5

so you say "nope." for the x=5

[midnight plank]

@cloud bronze Has your question been resolved?

[cloud bronze]

In this original example x=-2 is restricted

Right?

So always when you divide if the denominator can equal 0 then u restrict it

[midnight plank]

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[twilit field]

As each $X_i \sim N(0,1)$, the seqeunce $X_i$ converges to $N(0,1)$ and thus by cesaro means so does $\overline{X_n}$

[grand pond]

wai

[midnight plank]

@twilit field Has your question been resolved?

[ancient ridge]

I don't really see why you are trying to use Cesaro means here. F_n is normally distributed with mean 0 and variance 1/n then you can take the limit directly.

[twilit field]

oh right

stupid me

thanks

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[muted glen]

can someone help me with the second part

0.3 of 1.56 = 0.468

am i supposed to find (x/100)* (operating expense + cost of sales) = 0.468 ?

[midnight plank]

@muted glen Has your question been resolved?

[muted glen]

<@&286206848099549185>

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@muted glen Has your question been resolved?

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[supple crypt]

why the answer for 33 is not A

but D

[supple crypt]

but D

[fallow scarab]

#old-network for physics server

[midnight plank]

@supple crypt Has your question been resolved?

[clever sedge]

Because alpha particles have more Kinetic energy

They have much more mass

[supple crypt]

but more energy means less ionising right

[supple crypt]

ik that that's y I am wondering

y energy has to do with ionising

[clever sedge]

No, the opposite

[supple crypt]

there will be less time for the particle to ionsie

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[supple crypt]

.reopen

[midnight plank]

✅ Original question: #help-49 message

[supple crypt]

people there r non existent for answering

[clever sedge]

But Alpha particles are very slow

[supple crypt]

aah understood

i have to answer using the question

and not use any other theory knowledge

ty

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[clever sedge]

Yeah. Sorry, I am not really very well versed on this topic. Perhaps wait in the Physics server if anyone does answer 🤔

[supple crypt]

no nothing wrong in what u said

i thought alpha particles are only ionising bcz of their mass not bcz of their energy

in the question it says alpha particles travel faster than beta particle

[clever sedge]

The way I understand it is they have a lot of energy, are very slow and so they "deposit" this energy over a very short distance - low penetration and high ionizing potential

[clever sedge]

No it says beta particles have 20x the velocity of alpha particles

[supple crypt]

yea i got even more confused

i will ask my teacher

.close

[clever sedge]

👍

Best of luck and sorry 🙂

[supple crypt]

y sorry
u did help me
but I think the questions wording is not good

[clever sedge]

It's just that I am not used to leaving things unexplained or to ambiguity 🙃

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[last slate]

Woah I wanna learn this what class are you taking

[supple crypt]

it's an official past paper for igcse

[supple crypt]

but can we control the speed of emission of the particles

[west wadi]

I am pretty sure such particles are emitted by radioactive material which is kept where we want them. Basically the speed is a natural product of the radioactive decay so probably controlling the speed of emission is not possible.

But I think it can be done for charged particles via electromagnet. Asking your teacher is the best move since the information they give will be more reliable.

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[near geyser]

How do I check conditional convergent?

[small jasper]

Doesn’t the same argument of using the test for divergence apply?

Also, in the future, please use the ,rotate command

,rotate

[near geyser]

Given answer is oscillation finitely

[small jasper]

oscillates -> limit doesn’t exist

[nova yoke]

|a_n| -> 0 if and only if a_n -> 0

[near geyser]

What does this statement say?

[small jasper]

$\lim \limits_{n \to \infty} |a_n|=0$ if and only if $\lim \limits_{n \to \infty} a_n=0$

[grand pond]

Civil Service Pigeon

[small jasper]

The $\to$ represents the limit of the sequence (aka what the sequence approaches as $n$ tends to $\infty$)

[grand pond]

Civil Service Pigeon

[near geyser]

But in our question it tends to 1/2

So it is not conditional convergent too

Am I right?

[small jasper]

mhm

The absolute value of the sequence does not tend to zero, and thus the original sequence does not either. Hence, the series diverges by the test for divergence.

[near geyser]

How do we check oscillatory series?

I meant you said limit doesn't exist statement above

[small jasper]

Alternating is one of the easier cases since you can just take subsequences based on parity

[near geyser]

I didn't get it properly sorry

you want me to seperate positive and negative terms?

And then i have to see their limits?

[small jasper]

Yeah this is essentially what you’re doing since
$$(-1)^n=\begin{cases} 1, & n=2k \ -1, & n=2k+1 \end{cases}$$
where $k$ is an integer.

[grand pond]

Civil Service Pigeon

[small jasper]

So you’re effectively considering what happens as $n \to \infty$ when it’s odd and when it’s even

[grand pond]

Civil Service Pigeon

[small jasper]

these are the subsequences I was referring to earlier

(oh and parity just means whether it’s even or odd)

[near geyser]

We will get 1/2 or -1/2

Oscillates finitely

[small jasper]

mhm

[near geyser]

Am I right?

Thanks Civil

[small jasper]

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[near geyser]

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[rich mirage]

my question is in the s-t graph from t = 0 to t =0.224 the graph is concave up kindof, while from t = 0.224 to t = 0.447, its concave down how do i figure this out intutively and not have to rely on equations for parabolas

[fallow scarab]

You don't?

You just want to guess numbers and get the right answer?

[rich mirage]

no

[acoustic bane]

the s-t graph is the integral of the v-t graph

the rate of change of an integral is the original expression, the y-value

[rich mirage]

hmmm

how do i say this

oh well

thanks

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[lofty notch]

how do you find the fourier coefficient if the function is restrcted from 0 to pi/2?

[sharp coral]

it's not restricted

it just happens to be 0 elsewhere

[lofty notch]

whoops thank you so much i didnt see the abs value on the theta

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[sharp coral]

strangely it doesn't tell you what's going on for -pi/2 < theta < 0

which you might have to ask your teacher about

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[azure bolt]

10b

[acoustic bane]

how did you show 10a?

[azure bolt]

,, \textbf{Assume} \sec^2(\th) = \tan^2(\th) + 1
\begin{align*} \ &= \frac{\sin^2(\th)}{\cos^2(\th)} + 1
\ &= \frac{\sin^2(\th) +\cos^2(\th)}{\cos^2(\th)} \ &= \frac1{\cos^2(\th)}
\ &= \left(\frac1{\cos(\th)}\right)^2 \ &= \sec^2(\th) \ \text{LHS} &= \text{RHS} \end{align*}

[acoustic bane]

you started out with what you wanted to prove...

that's circular reasoning

[azure bolt]

what

they asked me to show that

lhs = rhs for proofs??

or am i confused with show that

no its a show that question

[acoustic bane]

tan^2 theta + 1 = sec^2 theta needs to be the last thing you write in your proof

[azure bolt]

but i wrote it here

i showed that tan^2 th + 1 = sec^2 th

[acoustic bane]

but you assumed it at the very beginning

[azure bolt]

show that is not the same as proof

[acoustic bane]

still

that's not a way to 'show that'

[azure bolt]

for show that its show that

[azure bolt]

i transposed my way through

wait

i dont think thats the right word

i algebraically reasoned

[acoustic bane]

you better write $\sec^2 \theta = \frac{1}{\cos^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta}$

[grand pond]

blitzchamp

[acoustic bane]

you have the right idea yes

[acoustic bane]

see how we're starting from the left-hand side, and working ourway to the right hand side?

[azure bolt]

yea

[acoustic bane]

it makes more sense to present your work this way

there's no structure to what you wrote before

[grand pond]

calvin

[azure bolt]

does that work

[acoustic bane]

no, you should never assume what you want to show

[azure bolt]

okay

so do i work backwards from sec squared

[acoustic bane]

yes, working backwards is the key idea

that can definitely be your rough work

[acoustic bane]

then yes you can reverse all your steps here

[azure bolt]

okay

thanks

[acoustic bane]

start from the bottom and go back line by line to the top

then that's a valid demonstration

np!

<@&268886789983436800>

[azure bolt]

can you help with 10b

i needed help with that bit

[acoustic bane]

okay, now we finally got to what I wanted to talk about

for which values of $\theta$ is $\sin^2 \theta + \cos^2 \theta = 1$ true?

[grand pond]

blitzchamp

[azure bolt]

all theta

[acoustic bane]

yep!

and then the other things you used are definitions

[azure bolt]

yeah

[acoustic bane]

e.g $\sec \theta = \frac{1}{\cos \theta}$ is true for all $\theta$

except when $\cos \theta = 0$, so that's the only exception

[grand pond]

blitzchamp

[azure bolt]

yep

[acoustic bane]

so 10a is true for all theta such that both sides are defined

that's probably the thing you didn't think about before

[azure bolt]

like domain

[acoustic bane]

exactly

except 90, 270 deg

[azure bolt]

270?

[acoustic bane]

yeah

[azure bolt]

why

[acoustic bane]

you could also think about it as when tan is undefined

[azure bolt]

oh

wait im confused

oh yeah

mb

not 180

thanks

[acoustic bane]

no worries!

[azure bolt]

should i always consider when the equation is defined

[acoustic bane]

yes

[azure bolt]

alr

[acoustic bane]

that's a good thing, to always pay attention to the domain of both sides

[azure bolt]

alr thank you

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[balmy cypress]

What is the difference in probability between the Poisson variable, the uniform variable and the binomial variable ?

[fresh sparrow]

different distributions

[balmy cypress]

i know

When are they used in exercises?

[fresh sparrow]

they are usually given

how a random variable is distributed

[balmy cypress]

what if it's not written?

[fresh sparrow]

show an example

[balmy cypress]

ok

A call center receives customer calls at an average rate of 3 calls per minute. Assume that:

calls occur one at a time, the chance of a call in a very small time interval is proportional to the length of the interval,

the numbers of calls occurring in disjoint time intervals are independent.

Answer the following:

What is the probability that exactly 5 calls arrive in a 2-minute interval?

What is the probability that at most 1 call arrives in the next 30 seconds?

What is the probability that no calls arrive in the next 20 seconds?

Given that at least one call arrives in the next minute, what is the probability that exactly one call arrives in that minute?

this is an example

[craggy pulsar]

Usually something like "How much time it takes for something to happen" is modelled by Poisson distribution

Binomail models thing that "happen or not" like coins etc

events that have two outcomes

@balmy cypress

[balmy cypress]

and uniform distribution?

[scenic wyvern]

uniform distr. sounds like any event that has some finite number of outcomes that are all equally likely

a single die roll is an example

[balmy cypress]

yes

[scenic wyvern]

a fair coin flip is also uniform

[balmy cypress]

👍

[balmy cypress]

but how long do you wait is the exponential distribution

[craggy pulsar]

Yeah you're right, I forgot much about my statistics course

but I remember these call center questions were modeled by Poisson

[scenic wyvern]

if the mean occurrance count per period of time is given it's usually Poisson

[balmy cypress]

Thanks

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[pure summit]

The circle of x^2+y^2+4x+4y-1=0 is cut by the line x-y+1=0 at two points A and B. If P is a variable point on the circle, then find the locus of the orthocentre of the triangle PAB?

[pure summit]

I know how to compute the points

How to take out orthocentre

[wanton glacier]

intersection of altitudes

[pure summit]

Any easier methods

It is very timetaking

[wanton glacier]

not that I can think of rn, maybe others could

but nothing to lose bashing it out

[pure summit]

No the points have surds in them

It is very timetaking

I tried already

[coral fern]

maybe try the euler line property? centroid divides the orthocentre and circumcentre in the ratio 2:1, you already know the circumcentre as the centre of the circle (-2, -2)

[vivid yoke]

that's cheating but yeah

[pure summit]

how is that cheating

[vivid yoke]

The whole proofs behind euler line

If you don't have to show that then it's fine

But if you do, it would be ass

[coral fern]

isnt it a standard result?

[vivid yoke]

It depends on the exam

[pure summit]

nope i dont have to show the proof

[vivid yoke]

We used to prove some properties of euler line on exam back then

[pure summit]

but i need a fast method

[vivid yoke]

well then this's fine

[coral fern]

ahh okay where i am its just taken as a standard property

[pure summit]

also another question

if i have an equation like

a/x^2 + b/x + c + d/x^(3/2) + e/x^(1/2)

is there any way i can simplify this without having to form a huge ahh eqn

[paper oak]

no

sorry to barge in. why is your name real finn wolfhard

[pure summit]

because im real finn wolfhard?

[paper oak]

how do I believe that?

[pure summit]

ion know

im live on snl

woo

[paper oak]

I have no idea what that is...but if you really are hats off

[grand pond]

scoob

[paper oak]

it still doesn't prove it.

[paper oak]

maybe by partition?

[pure summit]

rly think finn would be taking highschool maths and be miserable at it?

[paper oak]

no

maybe the miserable part. who knows. he is an actor and not a mathematician

only fans would think he would ace math or something

or maybe he has multiple passions on the other hand

good thing: there was a time around 2019 I was his huge fan. haha

[midnight plank]

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[near geyser]

I did not understand how 3/2 came

[twilit jetty]

you have to simplify this

simplify (3 an + 4) / (2 an + 3) to get the RHS

[near geyser]

But how?

[twilit jetty]

there are a few ways you can learn this

[near geyser]

What if I do this

[twilit jetty]

that doesnt really count

you still have an in the numerator

[near geyser]

Why?

[twilit jetty]

if an is still in the numerator, you still have the same problem

just doing + (3 + 2 an) or - (3 + 2 an) seems a bit spammy too

[near geyser]

I see

[twilit jetty]

surely you can pick

more specific multiples

of (3 + 2 an)

maybe one that would cancel out with 3 an

and would give you a 3/2 instead of just integers like 1 or 2

[near geyser]

So we have to take an in denominator

[twilit jetty]

I dont know what you just said

[near geyser]

We have to vanish an from up side

[near geyser]

How do we do?

[twilit jetty]

lets think about this

[near geyser]

Yeah sure

[twilit jetty]

it looks like you dont like the + # - # trick

now have you heard of the + # - # trick before?

[near geyser]

Yeah

[twilit jetty]

now couldnt you have just done + (3 + 2 an) - (3 + 2 an) here?

you didnt have to split apart 4 and 3 into those numbers

what if for example you had (4 - 3 an) / (3 + 2 an)?

instead of manually splitting apart and subtracting the numerator,

[twilit jetty]

you can do this

part of it then can be taken out as a 1

and the other part is just a calculation instead of a "reverse calculation"

the "reverse calculation" being "4 = 3 + ?" and "3 = 2 + ?" that you seem to have done

[near geyser]

4=3+1

[twilit jetty]

this does the same thing you did but with less effort and faster

[near geyser]

3=2+1

[twilit jetty]

it is faster to do 4 - 3 instead of "4 = 3 + ?"

since you should have the same method regardless of what numbers they give you in the numerator,

and the + # - # trick would work for fractions too:

,,\frac{\frac23+\frac{32}{183}a_n}{2+3a_n}

[grand pond]

mtt

[near geyser]

What???

[twilit jetty]

you dont have to think about subtracting the exact correct amount from these numbers

the + # - # trick does what your "manually splitting apart" idea fails

theyre identical, but one is faster

you should use it

[near geyser]

Actually i need help again ohhh no

What should I do for numerator)

[twilit jetty]

lets go over what I just said

[near geyser]

4+3an ...will be?

[twilit jetty]

popking

remember the purpose of this server

you have to figure out what to do

but Ill help you see what to do next

lets go over what we just said

[twilit jetty]

we agree now this trick is better than what you did, yes?

if you dont agree on this, the next steps will be very painful

wait

did you read what I was saying

I dont think you did at all

we gotta restart

here you did 4 = 3 + 1 and 3 = 2 + 1

[near geyser]

I guess yes

[twilit jetty]

you gotta be faster than that popking

[near geyser]

3+1+2an+an

[twilit jetty]

now that has a few problems

[near geyser]

Yes as u said

An did not vanish

[twilit jetty]

popking theres something important here

[twilit jetty]

even if an vanished, this method still isnt a good idea

mostly because theres a faster and better one

[near geyser]

our goal is to vanish an from numerator?

[twilit jetty]

wrong thinking

[near geyser]

Then?

[twilit jetty]

your goal is to finish the test without error and with understanding

now if youre doing "4 = 3 + ?" all the time instead of the simpler "4 - 3",

what happens if for example the problem started off by saying this?

[near geyser]

4-3=1

[twilit jetty]

hold on there popking

I am still typing

these things take time you know

I wasnt done yet

thats why it doesnt look like it makes sense yet

A sequence $\langle b_n\rangle$ is defined as $b_n=1;b_{n+1}=\frac{71+\frac{900}8b_n}{\frac89+\frac98b_n}$. Show that $b_n$ converges and find its limit.

youre not going to split apart one 8/9 + 9/8 bn at a time anymore with something like this

A sequence $\langle c_n\rangle$ is defined as $c_n=1;c_{n+1}=\frac{71+\frac{900}8ic_n}{\frac89-\frac98c_n}$. Show that $c_n$ converges and find its limit.

no matter what version of the problem theyre going to show you, you need to have a better way that can work for any problem

[grand pond]

mtt

[near geyser]

Yes. I am looking for it

[grand pond]

mtt

[twilit jetty]

let me show you one

for some reason you dont seem to realize that Im here

I could just tell you the method is + # - #

[near geyser]

I'm not getting it properly

[twilit jetty]

lets see it in action

[near geyser]

Any simple example?

[twilit jetty]

popking this is a conversation

you have to talk back for me to keep going

[near geyser]

Fine

[twilit jetty]

lets say we have something like this

now we need a fast way to pull out a lot of 8/9 + 9/8 bn

[twilit jetty]

now earlier you did 4 - 3

so heres a new idea

[twilit jetty]

rewrite 900/8 as 100 * 9/8

,,\frac{71+100\cdot\frac98b_n}{\frac89+\frac98b_n}

[grand pond]

mtt

[twilit jetty]

(that way its easier to see how many 9/8 bn we have in the numerator)

now around this point you wouldve done "71 = 100 * 8/9 + ?"

messy and can lead to mistakes under time pressure

[near geyser]

I guess i got it

[twilit jetty]

safer and simpler:

[near geyser]

We have to take out 9/8bn common

[twilit jetty]

not correct

you cant really do that here

,,\frac{71+100\cdot\frac98b_n\color{yellow}-100(\frac89+\frac98b_n)+100(\frac89+\frac98b_n)}{\frac89+\frac98b_n}

[grand pond]

mtt

[twilit jetty]

the safer and simpler way is to just do - 100 * denominator + 100 * denominator

first off, you change nothing here

so we know all we gotta do is simplify

now we do this

its the same calculation you wouldve done before

but this time its not in your head

its directly 71 - 100 * 8/9

instead of "71 = 100 * 8/9 + ?"

when written out, there is less chance of a mistake

and the method gets simpler as a result

[twilit jetty]

when you do this, what do you get?

[near geyser]

Tell me one thing

[twilit jetty]

please finish what I am saying

[near geyser]

We are removing an from numerator

[twilit jetty]

before you try and ask about it

[twilit jetty]

thats not correct

maybe what Im going to say next answers your quetsion

again we arent done yet here

maybe itll make more sense as I keep going

[near geyser]

Yes. I got it

[twilit jetty]

but right now, I just want you to subtract these two

@near geyser hello?

[near geyser]

Yes sir

[twilit jetty]

are you lost?

[near geyser]

Wait let me do it

Actually i am slow in reading english

[twilit jetty]

first time Ive heard that

ok Ill have to change how I type

you started trying this:

I say this wont work for other numbers

heres a better way:

now do this

before I continue

[near geyser]

71-800/9

[twilit jetty]

yea thats all

[near geyser]

Rest cancels out

[twilit jetty]

,,\frac{71-800/9+\color{yellow}100(\frac89+\frac98b_n)}{\frac89+\frac98b_n}

[grand pond]

mtt

[twilit jetty]

so we had - 100 (denominator) cancel out

the + 100 (denominator) is still there

,,100+\frac{71-800/9}{\frac89+\frac98b_n}

[grand pond]

mtt

[twilit jetty]

and just like that, no bn in the numerator

[twilit jetty]

done, really, in one step here

it will work for any fraction

because you just + # * (denominator) - # * (denominator)

then you choose # so that it cancels out

alr heres the other example

[twilit jetty]

this time, you tell me what the # is

[midnight plank]

@near geyser Has your question been resolved?

[twilit jetty]

so?

are you stuck?

@near geyser

[midnight plank]

@near geyser Has your question been resolved?

[midnight plank]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
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• Show your work, and if possible, explain where you are stuck.
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• Type the command .close to free the channel when you're done.
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[urban sentinel]

1500 * 0,90^x = 300 * 3,60^x

[urban sentinel]

First thing I thought to do was divide 1500/300

[lyric charm]

maybe good idea but bad phrasing

you shouldnt be trying to pluck numbers out of an equation and perform magic on them

do you maybe want to \textbf{divide both sides by 300}? then your equation will become $$\frac{1500}{300} \times 0.9^x = 3.6^x$$

[grand pond]

Ann

[urban sentinel]

That’s what I wanted to do and then make it like this

Lgx 0,9 = Lgx 3,6

Idk if that’s right tho

I know I need to bring that x down

[visual tiger]

Did you write log base x?

[urban sentinel]

No I haven’t touched it yet at all

Cause it felt wrong to divide by 300

[visual tiger]

No but like here "Lgx" means "log base x"?

[urban sentinel]

Ohhh

I think so yeah

[visual tiger]

Because that's not what's happening 😭

[urban sentinel]

Im just guessing with actions tbh im not that good with logarithms

[visual tiger]

Let's rewrite the equation the best way possible before even using logs

[visual tiger]

Using what Ann wrote, we now have $5 \times 0.9^x = 3.6^x$

Is there a way to regroup more alike terms?

[urban sentinel]

1500/300 is 5

[visual tiger]

Oops

[grand pond]

Rafilouyear2026

[urban sentinel]

I feel like I want to divide 3,6/0,9

[visual tiger]

That's late night math for you

[urban sentinel]

But can I do that

[visual tiger]

I mean, you can divide by whatever you want, as long as it makes sense

Can you write it down properly?

[urban sentinel]

Yes on paper you mean?

[visual tiger]

Yeah sure

[urban sentinel]

It’s kinda messy

[visual tiger]

You wrote down the correct thing so it doesn't matter

[urban sentinel]

Lg x 4= lg 5

[visual tiger]

Ok now let's calm down

[urban sentinel]

X= lg5/lg4

Lol

Im guessing

[visual tiger]

This is correct but what you wrote before is very wrong

[urban sentinel]

Lolll what did I do

[visual tiger]

So let's go back a little bit

[urban sentinel]

Yeah

[visual tiger]

You correctly simplified (3.6)^x/(0.9)^x = 4^x using exponent rules

So 4^x = 5

From there, we can apply a same function on both sides

We can apply the usual log, so "lg" as you write it

Btw is it base 10 or base e when you don't specify the base?

[urban sentinel]

Yes

[visual tiger]

I love it when my A or B question is answered with 'yes'

[urban sentinel]

Lol

[urban sentinel]

I mean the first is correct

The base 10

[visual tiger]

Ok

[lyric charm]

by the time you learn logs, you should be already familiar with "non-log" actions, namely:

• add something to both sides (also covers subtraction)
• multiply by something on both sides (also covers division)
• raise both sides to some power (also covers reciprocals and roots and shit)

symbolically, you should be able to perform the operations +a, -a, *a, /a and ^a on both sides where a is a constant or expression

[urban sentinel]

Oooh okay

[lyric charm]

thats sth you should know, perhaps all the way from linear eqs

e.g. x + 5 = 17 is solved by -5 both sides

[visual tiger]

More generally, if you have a function f, from "x = y" you can deduce "f(x) = f(y)"

[lyric charm]

once logs come into the picture, you also start caring about two other operations:

• a^ (apply the exponential function a^x to both sides)
• log_a (apply the logarithmic function log_a(x) to both sides)

note 2^ is very different from ^2

[urban sentinel]

Yess I know that sometimes I just bug with logarithms cause it’s new to me

[visual tiger]

Just like with basic linear equations, you would solve this by applying the function f = " ... minus 5"

[visual tiger]

You can use this statement with many other functions, such as squaring, multiplying by something... and now logs

[urban sentinel]

Oooh

[visual tiger]

So, if two quantities like 4^x and 5 are equal

Then their logs are equal

[urban sentinel]

Yeah

[visual tiger]

So, idk let's use the base 10 log for now

$\log(4^x) = \log 5$

[grand pond]

Rafilouyear2026

[visual tiger]

Makes sense?

[urban sentinel]

kinda

I thought the x would go down

[visual tiger]

Well not immediately

We have to do extra work after applying the log on both sides

Like, using log properties

[urban sentinel]

Ohhh okay

So now after we applied log we can bring down x

[visual tiger]

Do you remember how log behaves with exponents

[urban sentinel]

Somewhat yeah

[visual tiger]

$\log_b(a^x) = ?$

[grand pond]

Rafilouyear2026

[urban sentinel]

Doesn’t it become like I did before lg xa

[lyric charm]

log and not lg, and you mixed it up.

[urban sentinel]

They’re not the same?

You mean 10^lg ax

[visual tiger]

Let's try to clear up what you mean by that

$\log xa$? $\log_x a$? Something else?

[grand pond]

Rafilouyear2026

[urban sentinel]

Wait what’s the difference

[visual tiger]

Well, the first is the log of xa, with the usual base (base 10)

And the second is the log of a, base x

The base, if specified, is always written below as an index

[visual tiger]

In any case, both are wrong

Let's try to recall the correct formula with a specific case

[urban sentinel]

Okay

[visual tiger]

Do you remember the formula for $\log_b(xy)$?

[grand pond]

Rafilouyear2026

[visual tiger]

Log of a product

[lyric charm]

see THIS sort of misconception is why i am such a stickler for always putting brackets around inputs of logs.

[urban sentinel]

I don’t think so

[visual tiger]

We've got a long way to go

Do you at least remember what the exponential does to a sum?

$a^{x+y} = ?$

[grand pond]

Rafilouyear2026

[urban sentinel]

Wait do u mean these

I know these

[visual tiger]

Ok

[visual tiger]

So the exponential of a sum becomes (?)

[urban sentinel]

a^xy

Or

Omg 😭

[visual tiger]

First line (with a sign written wrong)

[urban sentinel]

Ohhh

A^x+a^y

[visual tiger]

💀 💀 💀 💀 💀

[visual tiger]

Thank you

[urban sentinel]

Clicked wrong my bad 🫩

[visual tiger]

So, when you input a sum (x+y) into an exponential

It outputs the product of exponentials a^x * a^y

[urban sentinel]

Yes

[visual tiger]

So, the exponential transforms sums into products, is that clear?

[urban sentinel]

Yes

[visual tiger]

So, since the logs are the reciprocals of exponentials

They're gonna do the reverse thing

They're gonna transform ... into ...

[urban sentinel]

What does reciprocals mean

[urban sentinel]

Products

[visual tiger]

Maybe you've called them inverse

[urban sentinel]

Im Swedish 🥲

[visual tiger]

Basically they revert things back to how they were

$f^{-1}$ is the inverse of $f$

[grand pond]

Rafilouyear2026

[visual tiger]

When you do one after the other, you end up back at the same place

[urban sentinel]

Yeah

[visual tiger]

$f^{-1}(f(x)) = x$

[grand pond]

Rafilouyear2026

[visual tiger]

And vice versa

[urban sentinel]

Yeah

[visual tiger]

So

If exponentials transform sums into products

And logs basically revert what exponentials do

Logs transform [...] into [...]

[urban sentinel]

Products to exponentials

Exponentials int productsZz

[visual tiger]

What exponentials do:

Sum -----> products

What would be the reverse of that

[urban sentinel]

The thing

That looks like what zikzak line

Idk what it’s called in English

[visual tiger]

Uhh we don't need that to understand what's happening

Say I went from a sum to a product, like below:

Sum ----> product

What would be the "other way around"?

[urban sentinel]

Product -> sum?

[visual tiger]

Yes!!!

So

If exponentials turn sums into products

Logs, being quite literally the "other way around"

Transform products into sums

$\log_b(xy) = \log_b(x) + \log_b(y)$

[grand pond]

Rafilouyear2026

[urban sentinel]

Ohhh

[visual tiger]

From a product "xy"

Applying the log leads to the sum of their respective logs

[urban sentinel]

Yeah

[visual tiger]

I could do this in fact for any number of terms in the product

If I had 3 terms: $\log_b(xyz) = \log_b(x) + \log_b(y)+\log_b(z)$

[grand pond]

Rafilouyear2026

[visual tiger]

And so on

[urban sentinel]

Okay I get it

[visual tiger]

So now

[visual tiger]

What happens if I apply this to x = y?

[urban sentinel]

Wait Wdym that doesn’t

Look the same

[visual tiger]

Mmh?

I don't understand what you meant by that

[urban sentinel]

Is it log(x)=log(y)

[visual tiger]

Well it would imply that

But take

The formula $\log_b(xy) = \log_b(x) + \log_b(y)$

[grand pond]

Rafilouyear2026

[visual tiger]

And now imagine that in that formula, x = y

[urban sentinel]

I don’t get it

[visual tiger]

We would then have $$\log_b(x\cdot x) = \log_b(x) + \log_b(x)...$$

[grand pond]

Rafilouyear2026

[visual tiger]

Since the formula is true for any x,y

I could replace y by whatever I want

Like, 72

[urban sentinel]

Whaaat

[visual tiger]

Or pi

Or x

[urban sentinel]

Im jus so confused

[visual tiger]

Ok, let's go through something else real quick, it might seem very weird but it seems it's heavily needed here

Imagine I have a class of students

And in that class of students, every student has brown eyes

Well, if I take "Dylan" who I know is in this class

I can say that Dylan has brown eyes

[urban sentinel]

Yes

[visual tiger]

Well

I have a list of real values x,y

And I know that for all those values

[visual tiger]

We have this nice equality

Well, if I take "62" who I know is in this list of values

I can say that the equality is true for y = 62

[urban sentinel]

Yeah

[visual tiger]

So

$\log_b(x\cdot 62) = \log_b(x) + \log_b(62)$

[grand pond]

Rafilouyear2026

[urban sentinel]

Yeah

[visual tiger]

But I could do the same thing I did here for other values

If I take "x" who I know is in this list of values

I can say the equality is true for y = x

So

$\log_b(x\cdot x) = \log_b(x) + \log_b(x)$

[grand pond]

Rafilouyear2026

[urban sentinel]

Uhhh

I mean I get what ur doing in the formula

[frank wolf]

x and y being different letters doesn't mean they have to be different values

[visual tiger]

And in the case that they are the same value, we have this

Abstraction is not always easy

[urban sentinel]

It’s fryinf my brain

[visual tiger]

Understanding that "x" and "y" are just placeholders is a big step

[urban sentinel]

Crazy that my teacher hasn’t explained any of this

[frank wolf]

what part is

[visual tiger]

Once you get that they are placeholders, as the name suggests, you can put anything inside of them

[urban sentinel]

That whole formula I’ve never see it before

[frank wolf]

wait which formula

[visual tiger]

Never seen this?

[urban sentinel]

No never

[frank wolf]

interesting

[visual tiger]

Ok... let's try another way then

I was planning to go through it afterwards, but it seems we can't do anything but that

[urban sentinel]

I have a huge exam tmr it’s like 4 hours long n my teacher was quite literally too lazy to teach us the functions we need in th calculator during the exam

She done said skip the questions

[visual tiger]

So, back to $4^x = 5$

[grand pond]

Rafilouyear2026

[urban sentinel]

Yes

[visual tiger]

Applying a random log will not necessarily help us

Since you don't have the formulas to deal with them

So, from 4^x

We want to retrieve x

Which means we need to get rid of the 4 at the base

Do you see which log to apply?

[urban sentinel]

Do you mean

10^4

[visual tiger]

?

No...

[urban sentinel]

…

[visual tiger]

Ok, do you remember when we said log are the reverse of exponentials

[urban sentinel]

Yes

[visual tiger]

So, if I take the exponential of x through some base

$b^x$

[grand pond]

Rafilouyear2026

[visual tiger]

And then I apply the log of the same base

$\log_b(b^x)$

[grand pond]

Rafilouyear2026

[visual tiger]

We end up like nothing happened

We end back at x

$\log_b(b^x) = x$

[grand pond]

Rafilouyear2026

[urban sentinel]

Yeah

[visual tiger]

Does that make sense?

[urban sentinel]

I think so

[visual tiger]

Ok

So If I have 4^x

How do I end back at x?

[urban sentinel]

Log(4^x)= x

Uh

[visual tiger]

You didn't write what was the base of the log

[urban sentinel]

Do I need to

[visual tiger]

If you don't write it, it'll be interpreted as base 10

And that doesn't work here

[urban sentinel]

I don’t get it at all cause I’ve never seen that

I just learned to do lg x 4= lg 5 an then get x alone

By dividing

By lg 4

[visual tiger]

Yeah but "lg x 4" isn't correct xddd

[clever sedge]

What

[visual tiger]

It's xlog(4)

The exponent goes in front of the log

But you don't know that because you haven't learnt the formulas for logs

[clever sedge]

So your teacher hasn't showed any identities to you?

[urban sentinel]

No she’s new

[clever sedge]

Hmm

[urban sentinel]

She doesn’t know what she’s doing

She uses her students as answer sheets

🥲

[visual tiger]

Just so we're clear, you don't have a sheet of formulas for log like the ones for a^...?

[urban sentinel]

Wdym

[clever sedge]

That can be frustrating. I get the teacher, it's difficult when you are new, but at the same time the students deserve the best ...

[urban sentinel]

This is my current formula paper

[clever sedge]

Well

Technically this can be solved with this

The 10^x = 7 line

But I assume

You haven't learned about bases of logarithms?

[frank wolf]

sometimes you work with logs over a single base rather than logs over general bases, to be fair

[visual tiger]

We have no choice than inspiring ourselves from this then

[urban sentinel]

What’s that

[clever sedge]

I see

Honestly

[visual tiger]

From $7^{2x} = 15$

[grand pond]

Rafilouyear2026

[frank wolf]

$\log_b(x)$ is defined to be $\frac{\log(x)}{\log(b)}$

[grand pond]

Pseudo (Cat theory #1 Fan)

[clever sedge]

My suggestion would be to read some material on logarithms and exponents

Because these are elementary topics you will always need

[urban sentinel]

My exam is tomorrow

[clever sedge]

Well

You can do it in a few hours

[urban sentinel]

I have ro be asleep in 4 hours and I got 50 ish questions left

To go through

Since my teacher sent out bad example questions before so now I’m taking the questions a different math teacher sent out

[visual tiger]

From $\textcolor{blue}{7}^{\textcolor{red}{2x}} = 15$ we deduce that $\textcolor{red}{2x}\log(\textcolor{blue}{7}) =\log (15)$

[grand pond]

Rafilouyear2026

[urban sentinel]

I understand what I did in that part of my sheet but idk how It correlates at all with this equation

[visual tiger]

When you apply the log, the exponent/power goes in front of the log

[urban sentinel]

Oh yeah

I did that

[visual tiger]

So

[urban sentinel]

7^2x= 2x log 7 we moved it down

[visual tiger]

From $\textcolor{blue}{4}^{\textcolor{red}{x}} = 5$ we deduce that...

[grand pond]

Rafilouyear2026

[urban sentinel]

It becomes

X log 4= log 5

[visual tiger]

Yessss xd

[urban sentinel]

So it was wrong before cause I did lg not log

[visual tiger]

well "lg" is how they write "log" in your class ig

So it's fine

[urban sentinel]

Yes

[visual tiger]

But whether you write it log or lg, you have to do it like that

[urban sentinel]

I’ll keep it in mind then

Not write lg x

x log

[visual tiger]

Both "x log4 = log5" and "x lg4 = lg5" are correct

But you were writing "lg x 4 = lg 5"

Which is incorrect

You were placing the x inside the log

[urban sentinel]

Cause of the x placement ?

In what situations does the x stay in the log then

[visual tiger]

A bit vague of a question

[visual tiger]

When dealing with equations like this

If you take the log

The power always comes in front of the log

[urban sentinel]

Okayyy

I had issues with another question

Where there’s z it’s supposed to be x

[visual tiger]

Another "Placeholder" related thing

If, in many exercises you've been doing, you've been trying to solve for x

On this one, they've decided to solve for "z"

[urban sentinel]

It’s the translator app bugging 🥲

It changes variables sometimes cause of the font

[visual tiger]

Ah ok

But don't be surprised if they truly use a new variable name one day

[urban sentinel]

Dw

I know

Im used to different variables

Im familiar with th word double root and real solutions

But

When it comes to answering questions

My teacher never brought it up

[visual tiger]

Have you heard of the discriminant formula for those

[urban sentinel]

How am i supposed to determine what q is gonna be for it to be double roots

[visual tiger]

$\Delta = b^2 - 4ac$

[grand pond]

Rafilouyear2026

[urban sentinel]

Show me idk the words in English

No I haven’t seen it

[visual tiger]

Huh

And... $x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$?

[grand pond]

Rafilouyear2026

[urban sentinel]

Aaaaaaa

Wait

This

Top one

[visual tiger]

They don't even use the full formula 😭 oh well

So, this formula gives you the exact solutions of $x^2 + px + q = 0$ depending on p and q

[grand pond]

Rafilouyear2026

[urban sentinel]

Usually q is a number though

Now it’s stayed a variable

[visual tiger]

q can be still seen as a number

We just don't know which one it is

[urban sentinel]

We need one side to be 0 thought right? But the 2 is there

It should be q-2

[visual tiger]

So let's make it 0 somehow

[urban sentinel]

But q is still a variable

[visual tiger]

It's fine, whether q is fixed or something that can change

The formula works nonetheless

[urban sentinel]

Do you want me to try it

[visual tiger]

Also, can you be more specific

Write the whole thing down I mean

[urban sentinel]

I mean to make one side 0 we need to deduct 2 from it, which we would’ve ben able to easily do and deduct 2 from q too if q was a number

Okay wait

I will

Warning my q looks like 9

[visual tiger]

Ouch

Not applied to the correct numbers

[urban sentinel]

Wdym

[visual tiger]

Lemme check but

Rewrite z² -10z -q = 2 into
z² + [...]z + [...] = 0

Specify what do I put into each box

[urban sentinel]

Wdym

[visual tiger]

Like, before applying the formula, you need to be able to tell me what are the "p" and "q"

That I'm gonna use in the formula

[urban sentinel]

Z^2+10z+q-2=0

[visual tiger]

So that's not what we have for starters

[urban sentinel]

Huuuh

[visual tiger]

we have z² -10z - q - 2 = 0

[urban sentinel]

Oh right q is negative

Forgot

And so was 10

Mb

[visual tiger]

q is negative has a minus sign in front*

[visual tiger]

So what is "p" in this formula

[urban sentinel]

-10

[visual tiger]

Ok

And... what is """"q""""

If you see what I mean

[urban sentinel]

Lol

Q

[visual tiger]

Ok you don't see what I mean

[urban sentinel]

Apparently not

Is it 2

-2

[visual tiger]

Not that either

[visual tiger]

I was talking about this """q""" here

We should use different letters because we're gonna mix up the two

[urban sentinel]

q..

q is q

[visual tiger]

We want something like x² + ax + b = 0

And right now we have z² - 10z - q - 2 = 0

So, the "a" is -10

And what is b?

[urban sentinel]

What

q

Or -2

Or what

[visual tiger]

It's neither... you have to account for every constant term

So b is "-q-2"

This whole thing

[urban sentinel]

Didn’t I write q as that

[visual tiger]

No, you used "q-2" here

[urban sentinel]

Ohhh

I forgot -

Sorry I’m a bit tired

But like I gotta finish this so

Tough it out

Shoukd I try again

But even if I did the -q

It wouldn’t just be

[visual tiger]

Yeah, with the correct coeffs this time

[urban sentinel]

27+q instead of 27-q

[visual tiger]

No, you wouldn't end up with 27

Try the math

[urban sentinel]

2 never merged with 25?

[visual tiger]

""merging"" 2 with 25 doesn't always give you 27... check the signs

[urban sentinel]

Wdym…

-(-q-2)

[visual tiger]

Do the math if you can't see it 😭 😭

[urban sentinel]

+q+2

[visual tiger]

Wait

Ok this is my bad on this one

It is 27+q under the root, I need to go to sleep

[urban sentinel]

🥲

So what after

I’m stuck there

[visual tiger]

Well, now you have "two" solutions for your equation, depending on q

[urban sentinel]

Yes

[visual tiger]

If you had a double root

That means +/- yield the same thing

So the thing inside the root must be 0

[urban sentinel]

Uhhh

[visual tiger]

Right now, you've written your two solutions as
"5 + Root"
and
"5 - Root"

[urban sentinel]

Yes

[visual tiger]

A "double root" means those two solutions are the same

[urban sentinel]

Yes