Question

How do i solve for x
2^3x = 4^x+2

Answer 1

is the problem\[2^{3x} = 4^{x+2}\]?

Answer 2

yes

Answer 3

in the future please use parentheses to avoid ambiguity:
2^(3x)=4^(x+2)
makes it a lot more clear to us...

to solve this problem you should rewrite 4 as a power of 2

any idea how you might do that?

Answer 4

2^4. How does that help me

Answer 5

4=2^2
and I will show you...

Answer 6

\[2^{3x}=4^{x+2}=(2^2)^{x+2}\]now remembering the exponent law\[(x^a)^b=x^{ab}\]we get\[2^{3x}=2^{2(x+2)}\]you should be able to take it from here

Answer 7

no, i dont know what to do

Answer 8

well you can at least simplify the exponent on the right\[2^{3x}=2^{2x+4}\]now is it clear what to do?

Answer 9

so solve for x, is that what you want me to do

Answer 10

\(I\) have no preference what you do frankly :P
but yeah, I'm pretty sure that's what they want

Answer 11

Dude, I have an F in algebra. Al i need to do is find x and I dont know how.

Answer 12

Well if you have no idea what to do from here you have missed a lot, so it's pretty hard to explain it quickly.

you should know that the inverse operation of exponentiation is taking the logarithm

both sides now have the same base, so you take the logarithm of both sides and solve for x

Answer 13

Others: do not give final answer

Answer 14

I dont need answer, I need the step on how to do this.

Answer 15

I think
you should ln(2^3x)=ln(4^x+2)

Answer 16

Cause i failed the logarithms test, and i need to know this so I can take the retakes and have an A again

Answer 17

ok if you want step-by-step that's good, but we need to go backwards a little bit for you to understand this problem

Answer 18

say we had\[2^{x+2}=2^{3x-5}\]since the inverse of raising 2 to some power is taking the log base 2 of that power, we take the log base 2 of both sides

Answer 19

2^(3x)=4^(x+2)

Create equivalent expressions in the equation that all have equal bases.

2^(3x)=2^2(x+2)

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

(3x)=2(x+2)

Multiply 2 by each term inside the parenthesis

3x=2x+4

Since 2x contains a variable to solve for, move it to the left-hand side of the equation by subtracting 2x from both sides.

3x-2x=4 Since 3x and -2x are like terms, add -2x and 3x to get x

Answer 20

\[\large\log_a(a^x)=x\]so\[\large\log_2(2^{x+2})=\log_2(2^{3x-5})\]which just leaves\[x+2=3x-5\]which you should be able to solve for x

this problem is the same, but with the caveat that we need to rewrite 4 as 2^2 to get all the bases the same

Answer 21

...I was continuing from my example above; I know that is not you exact problem

Answer 22

your*

Answer 23

k

Answer 24

can you do basic logs in your head?
like\[\log_3(27)\]for example?

Answer 25

yes. It equals 3

Answer 26

cause 3^3 = 27

Answer 27

ok, so do you understand the concept of the log (which is very helpful)
and have you commited to memory the properties\[\log(ab)=\log a+\log b\]\[\log\frac ab=\log a-\log b\]and\[\log(x^a)=a\log x\]?

Answer 28

so you do...*

Answer 29

yes

Answer 30

then we can solve this problem using that :)

Answer 31

so the tricky part here is rewriting the bases so they are all the same:\[2^{3x}=4^{x+2}\]\[2^{3x}=(2^2)^{x+2}\]\[2^{3x}=2^{2x+4}\]does this make sense so far?

Answer 32

yeah

Answer 33

so from here we use the third log property I wrote\[\log(a^b)=b\log a\]so here we get\[\log_2(2^{3x})=\log_2(2^{2x+4})\]\[3x\log_22=(2x+4)\log_22\]still with me?

Answer 34

yeah

Answer 35

great, now simplify:
what is\[\log_22\]?

Answer 36

it is 2/2

Answer 37

well my calculator says 0.3010 but i need fractions for answers

Answer 38

\[\large\log_22=?\]that is
"2 raised to what power equals 2 ? "

Answer 39

0.3010 but i need fractions

Answer 40

the heck with your calculator, your brain is faster
no it is not a decimal answer, this is log \(base\) \(two\)

Answer 41

2 to what power equals 2 ?
think about it...

Answer 42

1

Answer 43

yes, so all that gibberish above simplifies quite nicely :)
and what's left?

Answer 44

3x= 2x+4

Answer 45

nice, so x=?

Answer 46

x= 4

Answer 47

right, which is what buckethead said as well :)
go back and read his answer if mine is not sufficient and hopefully the two explanations together will help you see the concept

Answer 48

Thanks bro.

Answer 49

very welcome