How to simplify (e^ln(3x))(ln(e^2x))?

Question

Got a test and need to understand how to do it. Step by step will be great!

Answer 1

a^log[a](b) = b

log[a](a^b) = b



e^ln(3x) =

e^log[e](3x) =

3x



ln(e^(2x)) =

2x * ln(e) =

2x



(3x) * (2x) =

6x^2

Answer 2

Well you have to note that e^ln = ln(e)... they are the inverses of each other.



i.e. e^(ln(1)) = 1 and ln(e^1) = 1



so e^ln(3x)) becomes 3x

and ln(e^2x)) becomes 2x



so it simplifies to (3x)(2x) = 6x^2

Answer 3

e and ln are inverses of each other.



So raising e to the ln(#) just gives you the #



Similarly, taking the ln of e raised to some power (#) will also return the number



So e^ln(3x)=3x and ln(e^2x)=2x



Your answer is (2x)(3x)=6x^2

Answer 4

exp(ln(3x)) = 3x

ln(exp(2x)) = 2x

so the answer is 3x*2x = 6x^2

Answer 5

(e^ln(3x))(ln(e^2x)) = 3x(2x) = 6x^2

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Ideas: e^x and ln x are inverse pair.

Answer 6

You should consider that e^ln(f(x)) = f(x) and ln(e^(g(x))) = g(x)