Question

When and how do you use logarithms to solve exponential equations? give an example of an exponential equation that does not require logarithms to solve it and example of a problem that does require logarithms to solve it.

Answer 1

$\bf 5^{x+3}=\cfrac{1}{125}\implies 5^{x+3}=\cfrac{1}{5^3}\implies 5^{x+3}=5^{-3} \\\\\\ \textit{because the bases are the same, the exponents must also be the same} \\\\\\ x+3=-3\implies \boxed{x=-6}\\\\ -------------------------------\\\\ 3^x=4^{2x}\implies log(3^x)=log(4^{2x})\implies xlog(3)=(2x)log(4) \\\\\\ \cfrac{x}{2x}=\cfrac{log(4)}{log(3)}\implies \cfrac{1}{x}=\cfrac{log(4)}{log(3)}\implies \cfrac{log(3)}{log(4)}=x\implies \boxed{0.79248125\approx x}$