We're asked to find:

$\displaystyle \lim \limits_{t \to 0}\frac{5^t - 3^t}{t} $

The solution involves L'Hospital's Rule. In the solution manual, it displays the intermediate step as:

$\displaystyle \lim \limits_{t \to 0}\frac{5^t ln 5 - 3^t ln 3}{1} $

Which equals:

ln 5 - ln 3 = ln 3/5

Anyone have any clue how they (specifically) managed to get the $\displaystyle \frac{5^t ln 5 - 3^t ln 3}{1}$? Must be an algebraic trick that isn't occurring to me...?

Thanks in advance!