Question

Can someone help me prove this:

Answer 1

\[a ^{x} = e ^{(x)\ln a }\]

Answer 2

@AccessDenied

Answer 3

@iPwnBunnies

Answer 4

@hartnn

Answer 5

@ganeshie8

Answer 6

Did you have any ideas on how to approach this so far?

Answer 7

I only know the basic properties of natural logs.

for example: e^(lna) = a

Answer 8

Yup. How about the property for a power to a power?
\( (x^a)^b = x^{ab} \)

Answer 9

Yes. I am also familiar with that property. My question here is how to apply these properties to the formula provided.

Answer 10

I think we should try to convert the right-hand side into the left.
\( a^x = e^{x \ln a} \)
We have x times ln a. So how about we apply the power to a power property and rewrite it so that:
\( e^{x \ln a} = e^{(\ln a ) x} = \left( e^{\ln a} \right)^x \)

Answer 11

@AccessDenied Ohhhh! I knew that but this question dazzled me for while. Thanks a lot mate, I really appreciate your effort.

Answer 12

Glad to help! :)