# Math Help - Natural log rule explination pleaseee

1. ## Natural log rule explination pleaseee

Im really confused with the way my texbook explains the answer to this question:
lim as x approaches 0 of (5^x- 3^x)/x
So clearly thats impossible since there cant be a 0 in the denomintor so we use L'hospital's rule. Now the back of my textbook uses ln and this is where it got me confused:

lim (5^x- 3^x)/x = lim (5^xln5- 3^xln3)/1
I thought the general log rule was ln5^x = xln5 not 5^xln5 ???

2. Well, we can rewrite $5^x$ as $e^{x\cdot \ln 5}$. Now
$\frac{d}{dx} 5^x = \frac{d}{dx} e^{x\cdot \ln 5} = \ln 5 \cdot e^{x\cdot \ln 5} = \ln 5 \cdot 5^x$