# Thread: Where did the Log come from in this problem?

1. ## Where did the Log come from in this problem?

The following Problem:

$f(x)=5^{x^2+2x}$

Holds a derivative of:

$f'(x)=(2*log(5)*x+2*log(5))*5^{x^2+2*x}$

I need to know how this log came about.

2. $f(x)=5^{x^2+2x}$

$\ln f(x)=\ln \left(5^{x^2+2x}\right)$

$\ln f(x)=(x^2+2x) \ln 5$

$\frac{d}{dx}\left[\ln f(x)\right]=\frac{d}{dx}\left[(x^2+2x) \ln 5\right]$

$\frac{f'(x)}{f(x)}=(2x+2) \ln 5$

$f'(x)=f(x) (2x+2) \ln 5$

$f'(x)= 5^{x^2+2x} (2x+2) \ln 5$

Of course, when calculating a derivative like this, you wouldn't actually go through all these steps. Instead, you can just use the well known rule:

$\frac{d}{dx} a^{f(x)} = a^{f(x)} f'(x) \ln a$