log

[novel sleet]

how did you got the 2nd equation

,,x-1-3=0

[random estuary]

Renz

[sleek cedar]

divide both sides by 3

3 and 9

[plush sand]

Well, assuming x is not equal to zero, i agree with the second line.
After that, you can apply ln and get (x-1)ln(3) = 9

[novel sleet]

you cant divide $3^{x-1}$ by 3 and get x-1

[random estuary]

Renz

[novel sleet]

what you have to do is, factor out a 2x in the first step

,,2x\cdot3^{x-1}-18x=0

How would you factor that?

[random estuary]

Renz

[sleek cedar]

3^x-1 - 18x = 0

uhh bruhh

sorry

I mean

3^x-1 - 9 = 0

[novel sleet]

you divided by 2x

you only got to factor it out

it should be like this:

,,(2x)(3^{x-1}-9)=0

[random estuary]

Renz

[sleek cedar]

Yes but can’t we leave the 2x away?

Since we divided by 2x

And it would become 1

[novel sleet]

if you remove the 2x

you will miss a solution

so you have to include it

[sleek cedar]

Hmmm okay thanks did not know that

[novel sleet]

now, there are two factors being multiplied, and they equal 0

what can we do?

[sleek cedar]

Multiply both brackets?

2x times right bracket

[novel sleet]

nah

[sleek cedar]

Hmm but that woukd weird

Sorry idk

[novel sleet]

when two things are being multiplied

and they equal 0

it means either the first factor should be 0

or the second factor should be 0

meaning, you can equate each factor to 0

the first factor is 2x right?

[sleek cedar]

True

Aha so

[novel sleet]

and the second is $3^{x-1}-9$

[random estuary]

Renz

[sleek cedar]

2x should be to zero

[novel sleet]

yep

[sleek cedar]

Equal^

[novel sleet]

and $3^{x-1}-9$ must also equal 0

[random estuary]

Renz

[sleek cedar]

And obv the 2nd one aswell

[novel sleet]

yea

lets solve $2x=0$ first

[random estuary]

Renz

[novel sleet]

thats one very easy

[sleek cedar]

0

[novel sleet]

yea

[sleek cedar]

Divide by 2

[novel sleet]

it means x=0 is a soultion

[sleek cedar]

Both sides

[novel sleet]

to the original equation

now, lets do the other one

,,3^{x-1}-9=0

[random estuary]

Renz

[novel sleet]

what can we do

to the 9

[sleek cedar]

a

whut my wifi on ipad fell out

okay so

yeah

move to the 0

so wouuld be 9

[novel sleet]

correct

[sleek cedar]

Right side

So

3^x-1 = 9

[novel sleet]

,,3^{x-1}=9

[random estuary]

Renz

[sleek cedar]

yes

now we can do

[novel sleet]

now what

[sleek cedar]

hmm nvm ig

I would say

nahh nvm sorry

[novel sleet]

logs?

[sleek cedar]

divide by 3 but thats obv not possible

[novel sleet]

rewriting the base?

[sleek cedar]

Might be, I find it them hard the ln and e

[novel sleet]

ah

[sleek cedar]

idk srry

I hate them for now

[novel sleet]

lets go with rewriting the base i guess

[sleek cedar]

but ill get them asap

[novel sleet]

what do you notice with 9

[sleek cedar]

base as log base or ln?

Or doesn't matter?

[novel sleet]

eh doesnt matter

ill teach you the log method later

[sleek cedar]

Sure thanks

[novel sleet]

but for now, lets do rewriting the base

[sleek cedar]

Okay

[novel sleet]

what can we do with 9

,,3^{x-1}=9

[random estuary]

Renz

[sleek cedar]

write it as

[novel sleet]

our goal is to make the bases the same

[sleek cedar]

log3(2)

[novel sleet]

nope

[sleek cedar]

noo bruh

mb

[novel sleet]

9 is a perfect square

meaning?

[sleek cedar]

true

[novel sleet]

how would you rewrite it

[sleek cedar]

aha

3^2

[novel sleet]

Yea

thats correct

,,3^{x-1}=3^2

[random estuary]

Renz

[sleek cedar]

hmm

x-1 = 2

[novel sleet]

our bases are now the same!

[sleek cedar]

x = 3

[novel sleet]

,,x-1=2

[random estuary]

Renz

[novel sleet]

now solving for x

[sleek cedar]

x = 3

[novel sleet]

correct

[sleek cedar]

damm yeah that's really great

wiw

wow

[novel sleet]

meaning, x=3 is also a solution

[sleek cedar]

thanks a lot for the explanation!!

[novel sleet]

as well as x=0

[sleek cedar]

true

[novel sleet]

alright, lets do the log method

[sleek cedar]

Do we also have to check?

Like if it's possible

[novel sleet]

yea you can check

[sleek cedar]

because I know with root equation we do have to check

[novel sleet]

just substitue x=0 and x=3 in the original equation

it should give you 0

[sleek cedar]

yeah true it's correct

Nice wow yes thanks a lot

[novel sleet]

,,2x\cdot3^{x-1}-18x=0

[random estuary]

Renz

[sleek cedar]

sure ty leggo

[novel sleet]

alright lets start from here

,,3^{x-1}=9

from here

[random estuary]

Renz

[novel sleet]

to use the "log" method, you are esentially taking the logarithm of both sides, in order to bring the exponent down

which base is being raised to an exponent with a variable?

[sleek cedar]

left

side

[novel sleet]

and whats the base?

whats the base of the left side

[sleek cedar]

uhmm log3(x-1)

3 is the base

[novel sleet]

3 is the base of the exponent

[sleek cedar]

hmmm

[novel sleet]

meaning, you wanna take the log base 3 of both sides

like this

,,log_3(3^{x-1})=log_3(9)

[random estuary]

Renz

[sleek cedar]

Bruh okay nice hh. I honestly would have thought to leave 3 out of the brackets

Since we use it as our base

[novel sleet]

nah

we keep it in

[sleek cedar]

But this is great

[novel sleet]

because now, we can use the log property

,,log(a^b)=blog(a)

[random estuary]

Renz

[novel sleet]

if we are taking a logarithm of a number being raised to an exponent

we can bring the exponent down as a coefficeint

look at b

b became a coefficient

[sleek cedar]

Yes true

[novel sleet]

alright, lets apply that

,,log_3(3^{x-1})=log_3(9)

[random estuary]

Renz

[sleek cedar]

So

[novel sleet]

what do you think is the exponent inside the parenthesis on the left

[sleek cedar]

x - 1

[novel sleet]

yea

[sleek cedar]

And 3 3 cancel out

[novel sleet]

so (x-1) becomes the coefficient

,,(x-1)log_3(3)=log_3(9)

[random estuary]

Renz

[sleek cedar]

Okay nice, one question

[novel sleet]

and as you said log base 3 of 3 is cancelled

[sleek cedar]

Aha hes true

[novel sleet]

go on

[sleek cedar]

Yes

No it was about that

[novel sleet]

ahh

just remember this property:

,,log_a(a) = 1

[random estuary]

Renz

[sleek cedar]

You are doing a great job by explaining step by step

[novel sleet]

:D

[sleek cedar]

True

[novel sleet]

alright

so log base 3 of 3 is just 1

we can remove that

[sleek cedar]

True

Yes

[novel sleet]

,,(x-1)log_3(3)=log_3(9)

[random estuary]

Renz

[novel sleet]

,,(x-1)=log_3(9)

[random estuary]

Renz

[sleek cedar]

So now we know

[novel sleet]

now, lets do things on the right side

[sleek cedar]

Right side is 2

[novel sleet]

correct

,,(x-1)=2

[sleek cedar]

x - 1 = 2

[random estuary]

Renz

[novel sleet]

and we can see we are in the same spot as earlier

[sleek cedar]

Yeahh woow very nice

True

[novel sleet]

x=3

[sleek cedar]

Dammm thats really nice

For real

But

Which one do you consider?

Like for me

[novel sleet]

like which one to use?

[sleek cedar]

Im a newb in logs

[sleek cedar]

Yeah

[novel sleet]

ah

for problems like that

rewriting the base is easier

however, rewriting the base may not be applicable the most exponential equations, as not all equations will give you a perfect square, or a perfect cube, etc.

so you gotta learn logarithms as well

learning logarithms will help you solve most exponential equations

even if there is no perfect square or perfect power

[sleek cedar]

Okay thanks, idk if you wanna resoind on this one

It’s an exercise I have left which cane pretty early in the chapter

[novel sleet]

hmm?

[sleek cedar]

Shoukd be easy but im cinfused w the answer

log15(5) + log15(3)

I know

We normally should use the multiplication

But in this case it seems different

[novel sleet]

hmm?, whats with that

you gotta use the logarithm properties

for instance

[sleek cedar]

Yes

What would you say is the answer?

I would say

[novel sleet]

,,log_a(x)+log_a(y)=log_a(xy)

[random estuary]

Renz

[sleek cedar]

Omg

Bruh

Sorry

Im so blind

It would obv be 1

Bruuuhhh

[novel sleet]

yea

[sleek cedar]

Sorry r

[novel sleet]

you gotta remember the properties

what happens when you add two logs of the same base

what happens when you subtract two logs of the same base

[sleek cedar]

fraction

[novel sleet]

yea

[sleek cedar]

the thing is we will have a cheat sheet handed out to us

[novel sleet]

,,log_a(x)-log_a(y)=log_a(\frac{x}{y})

[random estuary]

Renz

[sleek cedar]

with the 4 log rules

[novel sleet]

oo

[sleek cedar]

but as you make them, they;ll get obvious

[novel sleet]

yea, i would suggest just practicing

[sleek cedar]

Wait are you from the Netherlands?

[novel sleet]

its easy to remember

nah

[sleek cedar]

Oh okay srry xD

it sounded as a Dutch name

or Deutsch

[novel sleet]

ah xD

[sleek cedar]

Idk if you wanna help me w this one as well

got it on my last exam

but will have a reentrance exam next friday

[novel sleet]

let me see

[sleek cedar]

yes 1m gotta login

Idk if you are familiar with differentiation? Most likely..

[novel sleet]

differentiation?

leme see

[sleek cedar]

f(x) = 4x^2 for example

and then

[novel sleet]

i have not taken any actual calculus classes yet, but i do know things about it

[sleek cedar]

f'(x) = 8x

[novel sleet]

so you wanna differentiate that function?

yea

[sleek cedar]

Ahh know xD it was just example

[novel sleet]

its called the power rule for derivatives

[sleek cedar]

It was this one

I found it hard

[novel sleet]

ahh

[sleek cedar]

idk if you might know

we could write

[novel sleet]

alright lets take it step by step

[sleek cedar]

I think

as I think it by now

as

x3/5

• 5x^1/5

[novel sleet]

,,f(x)=\sqrt[5]{x^3+5x-4}

[random estuary]

Renz

[sleek cedar]

wait ill go back on my ipad to show you it's cleaner than laptop

[novel sleet]

you have this right?

[sleek cedar]

yes

true

[novel sleet]

alright, what we can do is

make the 5th root into an exponent

which is 1/5

[sour apex]

yes

[sleek cedar]

true

[novel sleet]

ive seen you typed it

alright

now we have

[sleek cedar]

hi Shaquille 👋

[novel sleet]

,,f(x)=(x^3+5x-4)^{\frac{1}{5}}

[random estuary]

Renz

[sleek cedar]

yes true

but the thingy is

[sour apex]

what is the question?

[novel sleet]

whats with it

[sleek cedar]

this was my real entrance exam wuestion

[sour apex]

differentiation?

[sleek cedar]

question

yeah

sorry

I know how to differentiate

[novel sleet]

do you know the power rule and the chain rule?

[sleek cedar]

but I panic when I saw this

[sleek cedar]

yes

we get these as well

[sour apex]

dont panic

power bring down power -1

then chain rule

[sleek cedar]

sorry

[novel sleet]

yea that sums it up

[sleek cedar]

wdym

I had 0 out of 5 mars 😦

[novel sleet]

,,f(x)=(x^3+5x-4)^{\frac{1}{5}}

[sleek cedar]

marks*

[random estuary]

Renz

[sleek cedar]

didnt do it

😦

[novel sleet]

alright we have this right?

[sleek cedar]

trye

[novel sleet]

we bring the power down to the coefficient

since thats what the power rule tells us

,,f(x)=\frac{1}{5}(x^3+5x-4)^{\frac{1}{5}}

[sour apex]

1/5(eq)^-1/5

[random estuary]

Renz

[sleek cedar]

sorru but which power rule

[novel sleet]

now, the power rule also tells us to subtract 1 from the exponent

[novel sleet]

this on

[novel sleet]

,,\frac{d}{dx}x^n=nx^{n-1}

[random estuary]

Renz

[sour apex]

yes

[novel sleet]

if you are doing calculus

[sleek cedar]

urghhh such a new layout okay so

[novel sleet]

people call it as power rule

[sour apex]

yes

[novel sleet]

but specifically, its called power rule for derivatives

[sleek cedar]

okay thanks

aha you mean the general rule?

Yes I see it now

[novel sleet]

because "power rule' alone may refer to many things

theres power rule for exponents

power rule for integration

etc.

[sleek cedar]

True

So after this step

[novel sleet]

we bring 1/5 down

,,f(x)=\frac{1}{5}(x^3+5x-4)^{\frac{1}{5}}

[random estuary]

Renz

[sleek cedar]

Okay so bringing down means multiply ig

[novel sleet]

then, we want to subtract 1 from 1/5 in the exponent

yea

whats

,,\frac{1}{5}-1

[random estuary]

Renz

[sleek cedar]

• 4/5

Bruh

[novel sleet]

ye its -4/5

[sleek cedar]

Minus 4/5

Yeah xd

[novel sleet]

im assuming thats a typo lol

[sleek cedar]

It did it automatically

[novel sleet]

,,f(x)=\frac{1}{5}(x^3+5x-4)^{\frac{1}{5}-1}

[random estuary]

Renz

[novel sleet]

which is:

,,f(x)=\frac{1}{5}(x^3+5x-4)^{-\frac{4}{5}}

[random estuary]

Renz

[novel sleet]

since we used a power rule on the whole thing

specifically on $x^3+5x-4$

[random estuary]

Renz

[novel sleet]

according to the chain rule,

[novel sleet]

we gotta multiply this by the derivative of the inside

whats the derivative of the inside?

of $x^3+5x-4$

[random estuary]

Renz

[sleek cedar]

3x^2 + 5

[novel sleet]

correct

so we just take this

[sleek cedar]

And instead of +

[novel sleet]

,,f(x)=\frac{1}{5}(x^3+5x-4)^{-\frac{4}{5}}

[random estuary]

Renz

[sleek cedar]

We multiply

Nvm ig

[novel sleet]

and multiply it by $3x^2+5$

[random estuary]

Renz

[sleek cedar]

Whaaaa

[novel sleet]

like this:

,,f(x)=\frac{1}{5}(x^3+5x-4)^{-\frac{4}{5}}(3x^2+5)

[random estuary]

Renz

[sleek cedar]

Aha

This is kinda confusing but ket me try

Broo noo

Nvm

Idk where to start

[novel sleet]

are you trying to simplify it?

[sleek cedar]

Idk, I guess xD

This doesn’t seem as the final answer for me

[novel sleet]

yea, answers should usually be simplified

alright

we have a negative exponent right?

lets remove that

its ugly xD

[sleek cedar]

Truee xDD

Nirmslly

[novel sleet]

we have a negative exponent

so where do we put it?

[sleek cedar]

We can switch fraction

[novel sleet]

yea

we put it with 5

in the denomintaor

like this:

,,\frac{3x^2+5}{5(x^3+5x-4)^{\frac{4}{5}}}

[sleek cedar]

Just erase it😍✏️

[sour apex]

yikes i needa go back to kindergarten

[sleek cedar]

Whoopsie

[sour apex]

i didnt know 1/5-1=-4/5

[sleek cedar]

Bruuh whaat

[novel sleet]

xD

[sour apex]

pretty sure u gotta remove the -

[novel sleet]

oh right

forgot

[random estuary]

Renz

[novel sleet]

now much better

i just copy pasted the things lol

[sour apex]

and do it in the form of root 5 square 4

[novel sleet]

yea

that ^

answer choices are usually in radicals instead of fractional exponents

so we do

[sour apex]

how do u use this texti

[novel sleet]

,,\frac{3x^2+5}{5\sqrt[5]{(x^3+5x-4)^4}}

[random estuary]

Renz

[sour apex]

,,\frac{3x^2+5}{5\sqrt[5]{(x^3+5x-4)^4}}

[novel sleet]

the bot thingy right

[random estuary]

Shaquille Oatmeal

[sour apex]

yeah

[novel sleet]

type in ,,x+2

,, means you are entering math mode

[sour apex]

,,x+2

[random estuary]

Renz

Shaquille Oatmeal

[sour apex]

,,x^2

[novel sleet]

or you can do dollar signs like this $y=mx+b$

[random estuary]

Shaquille Oatmeal

Renz

[sour apex]

interesting

[novel sleet]

search Latex equations on google

theres some syntax you can follow

[sour apex]

alright cool 👍

[novel sleet]

for example for a fraction, you do \frac{numerator}{denominator}

like $\frac{N}{D}$

[random estuary]

Renz

[sour apex]

yikes looks confusing

[novel sleet]

theres #latex-help and #latex-testing

yea

but you will get used to it

[sour apex]

alright

[sleek cedar]

https://tenor.com/view/thinking-smiling-day-dreaming-gif-14873557

[novel sleet]

@sleek cedar does this look more familiar

in the answer choices

,,\frac{3x^2+5}{5\sqrt[5]{(x^3+5x-4)^4}}

[sleek cedar]

No

[random estuary]

Renz

[novel sleet]

hmm

[sleek cedar]

Atleast we had a familiar one earlier on

[novel sleet]

what are the answer choices

[sleek cedar]

We did too much i gues xD

[novel sleet]

cus thats usually the accepted answer

[sleek cedar]

Let me makw a pic

There are 2 ways that leads to 1 solution

[sleek cedar]

Correct

[novel sleet]

ah lol

the answer key didnt simplified it

[sleek cedar]

Damm that would made me pass 1st time

Yeah I remember you asking if it shoukd’ve been multiplied

But yeah that’s also a thing im unsure about

When to simplify or not

[novel sleet]

is that a multiple choice question'

if it is, you should be seeing your answer as you simplify

if you teacher is gonna check it manually, then you should still recieve marks even if you simplified

but it still depends on the teacher's instruction

[sleek cedar]

Nope, just solving it. No multiple choices are given

[sleek cedar]

That's true. I might stick to that.

[sleek cedar]

.close