# Thread: Derivative of a log function to base e^x

1. ## Derivative of a log function to base e^x

Unsure if what I am doing with this function is correct;
I need to differentiate the following function,

$f(x) = \log_{e^x}(\sin x)$
I have used the change of base property
$\log_{b}x = \frac{\log_{a}x}{\log_{a}b}$
Thereofre;
$f(x)=\frac {\ln(\sin x)} {\ln e^x} = \frac {\ln(\sin x)} {x}$

2. Originally Posted by Robb
Unsure if what I am doing with this function is correct;
I need to differentiate the following function,

$f(x) = \log_{e^x}(\sin x)$
I have used the change of base property
$\log_{b}x = \frac{\log_{a}x}{\log_{a}b}$
Thereofre;
$f(x)=\frac {\ln(\sin x)} {\ln e^x} = \frac {\ln(\sin x)} {x}$
So your base is itself a variable? Wow, that is tricky! Yes, it looks to me like you have it reduced properly. Now just use the quotient rule and chain rule.