Derivatives of ex and ln x

Question

An example in my textbook goes as follows:

f(x)=ln e^{3x     = 3x}

f'(x) = 3

 

I understand that the derivative of 3x is 3, that the derivative of e^x is e^x, ,and that the derivative of ln x is 1 over x.

 

However, I still don't completely understand how ln e^3x can be simplified to 3x. Could someone please help clarify this?

 

Answer 1

The natural log (ln) is the inverse of the exponent function, e^x.  So by definition, ln(e^3x) = 3x.

 

Hence f(x) = 3x, so df(x)/dx = 3.

 

You may want to review your log identities:

 

http://www.math.com/tables/algebra/exponents.htm

Answer 2

It's based on the definition of a logarithm.

 

In words, the logarithm of a number, in a certain base, is the power you have to raise the base to in order to get the number.

 

So ln e^3x is the number you have to raise e to in order to get e^3x.  Well, what number do you have to raise e to in order to get e^3x ? By definition, it's just 3x!