Exponentials & logarithms

Exponentials and logarithms are opposite operations. The function $a^x$ (with $a>0$, $a\ne 1$) raises a base $a$ to a power $x$, while the logarithm ${\log }_{a}y$ asks “to what exponent must $a$ be raised to get $y$?”. Because they undo each other, ${\log }_a(a^x)=x$ and $a^{{\log }_{a}y}=y$. This lets us turn equations with powers into linear ones by taking logs, and turn log equations into powers by exponentiating. When the same base appears on both sides, you can often rewrite everything with that base and compare exponents directly. Remember the change‑of‑base rule ${\log }_{b}c=\frac{{\log }_{k}c}{{\log }_{k}b}$ for any convenient base $k$, such as $10$ or $e$, which is useful when the given base is not convenient.