Deriving Relationships: Given a^x = e^x ln(a), derive e^x, 10^x, ln(x), log_10(x)

[chillan]

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?
I have re-written the equation to a^x = (e^ln(a))^x

What are some important rules that I should start from? (chain rule, etc)

 

[Dr.Peterson]

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?
I have re-written the equation to a^x = (e^ln(a))^x

What are some important rules that I should start from? (chain rule, etc)

We'll need to clarify some things.

First, be careful with notation. You meant a^x = e^{xlna}, where I've added braces (which could just be parentheses) to show that it is not just x that is in the exponent.

Second, I'm not sure how you are using the word "derive". Your mention of the chain rule suggests that you are talking about taking the derivative (differentiating), which some people seem to call "deriving" (wrongly, I think, but perhaps not in their culture); but the question itself doesn't read that way to me.

I would interpret "derive the relationship between e^x and 10^x" as saying just to apply the identity to the base 10, namely that 10^x = e^{x ln10}, taking a = 10. That is a relationship between the functions!

The relationship between ln x and log x would be a form of the "change of base formula", which can be proved (derived) in several ways, one of them being your identity.

Is there any context to make it clear what is intended? Can you look up the answer to see what they meant? Otherwise, if you can, I'd just ignore the question as being poorly stated and not worth my effort in reviewing the subject.