Help with deriving relationships starting with the identity a^x = e^xlna

[CTil]

Hi there - it has been quite a long time since I took Calculus. I am trying to brush up and understand where to start with this question:

Starting with the identity a^x = e^xlna, derive the relationships between (a) e^x and 10^x; (b) ln x and log x. Note: log x = log10 x unless otherwise specified.

I know how the a^x was derived, but I'm honestly not sure what a) is asking? Is the 10^x regarding logarithm?
I have re-written the equation to a^x = (e^ln(a))^x

 

[Charles Link]

Can you write $10$ in the form $e^b$? What is $b$? That should help get you started.

 

[CTil]

here's where I am at...

f(x) = 10^x; 10^x = e^xln10; f′(x) = e^xln10 (ln 10) = 10^x ln 10
then from there I got 10^x 2.303 log 10...

 

[Charles Link]

The second part (b), in my opinion, is kind of a clumsy one in using the equation that is given. If you let $N=a^x=e^{x \ln{a}}$, then you can write, with $a=10$, that $x=\log_{10}(N)$ , but also $\ln{N}=x \ln{10}$. (Without introducing $N$, I think it is more difficult). Connect these last two expressions that each have an $x$. $\\$ Finally replace $N$ with an $x$. ($N$ represents an arbitrary number in this last equation that you obtain, so you can replace it with any letter you choose. This new $x$ is, of course, totally unrelated to the first $x$).

 

[CTil]

So if I am following you correctly, ln N/ln 10 = log10 (N) or ln X / ln 10 = log 10 X ?

 

[Charles Link]

So if I am following you correctly, ln N/ln 10 = log10 (N) or ln X / ln 10 = log 10 X ?

Correct.