lne equation

[gusty jay]

Hello, can anyone explain this equation? Thanks in advance.

[plush geyser]

its you again

what have you tried

[gusty jay]

Hehe, hi

Well we obviously can factor out 3^x

But the thing that embarrasses me, is the thing left after factoring out

I would say after factoring out 3^x we will have (1-x) in the brackets

[plush geyser]

ahh

use this property

so you dont get confused

,,x^{m+n}=x^mx^n

[runic compass]

Renz

[plush geyser]

try rewriting $3^{x+1}$ first

[runic compass]

Renz

[gusty jay]

3^x + 3

?

[plush geyser]

so we have 3 as our base right

and in the exoonents, we have x+1

[gusty jay]

True

[plush geyser]

according to this property we can rewrite it as

,,3^{x+1}=3^x\cdot 3^1

[runic compass]

Renz

[gusty jay]

Ah true yes

Mybad

But can’t we ignore the power 1?

Ah but this makes sense now

See:

[plush geyser]

yea we can ignore that after

ive written it down so you would know how we broke down "x+1"

[gusty jay]

Yeah true thank u

Okay but can we write the other 3^x as 3x as well?

[plush geyser]

huh why it became 3x

its $3^x$ btw

[gusty jay]

Oh bro

[runic compass]

Renz

[gusty jay]

Nvm

Yeah

Sorry

Its still morning xD

[plush geyser]

ah

now, what you notice with the 2 terms

do they have somethin in common you can factor

[gusty jay]

https://tenor.com/view/high-five-patrick-star-spongebob-squarepants-the-patrick-star-show-yes-gif-22559195

[gusty jay]

3^x. 🔥

[plush geyser]

yea

thats right

now, how about $3^x=0$

[runic compass]

Renz

[gusty jay]

That is not possible

Because 3^x always has a real positive number

ChatGPT told me yesterday

But yeah honestly dk what it means

[plush geyser]

ah

alright

its because theres no such exponent that will give you 0

and if you looked at the graph of a simple exponential function

such as 2^x

you will see that there is a horizontal asymptote at x=0

as x approaches negative infinity

meaning, the function 2^x or any exponential function in the form a^x will never touch the x-axis

thus, there will be no solutions to $a^x=0$

[runic compass]

Renz

[gusty jay]

True

Well

Yeah its true

Since power 0 equals 1

Nice

Can we approach

For more equations? Like when to know we can use this?

[plush geyser]

ah

leme make you one

[gusty jay]

Yes please

Thanks

[plush geyser]

lets do this ig

[gusty jay]

Sure

Thanks

[plush geyser]

,,e^{2x}\cdot e^x = 6

[runic compass]

Renz

[gusty jay]

I would say use the formula

[plush geyser]

what do you think we can do with the e's

[gusty jay]

I would say use the formula

[plush geyser]

yea

[gusty jay]

So we get

[plush geyser]

so if we are multiplying exponents with the same base, we have to add the powers

[gusty jay]

Uhmmm okay..?

[plush geyser]

we should not be cancelling them

[gusty jay]

Okay sorry

So

[plush geyser]

remember this

,,x\cdot x \x^1\cdot x^1 \ x^{1+1} \ x^2

[runic compass]

Renz

[plush geyser]

to multiply x with x

we have to add their powers

[gusty jay]

Thanks I saved it

[plush geyser]

since they share a common base

[gusty jay]

So

[plush geyser]

its the same in our case

so we do

,,e^{2x} \cdot e^x = 6

[runic compass]

Renz

[plush geyser]

,,e^{2x+x} = 6

[runic compass]

Renz

[gusty jay]

Huh im confused

It stays the same

No

e^3x

[gusty jay]

This was honestly my first idea, since I saw it in the book

But idk looks weird

For ke

[plush geyser]

ye

it becomes $e^{3x}$

[runic compass]

Renz

[gusty jay]

3x = lne(6)

[plush geyser]

correct

and finally, divide by 3

[gusty jay]

Bruh what will happen than

Ehmm

Ehmm

So

[plush geyser]

,,3x=ln(6)

[runic compass]

Renz

[plush geyser]

,,x=\frac{ln(6)}{3}

[runic compass]

Renz

[plush geyser]

now that cannot be further simplified

you have to use a calculator to get the numerical value of that

[gusty jay]

Oh damm

Did not know we can write it like thay

Bumm correct🔥🔥

Thanks

[gusty jay]

I was thinking more complex with dividing by 3 idk why

Is there any tips to know when to keep it simple?

[plush geyser]

wdym by "keeping it simple"

[gusty jay]

Like in what cases to simplify?

Or not

[plush geyser]

well

ln6 cannot be further simplified without using a calculator

usually simplifying things involve factoring things out

[gusty jay]

Yeah okay true. Well thanks a lot!

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