Advanced Algebra

$\log_b(xy) = \log_b x+\log_b y$, $\log_b(x^n) = n\log_b x$, and $\log_b x = \dfrac{\log x}{\log b}$.

In plain terms. Logs turn multiplication into addition and pull exponents out front, which untangles equations with a variable in the exponent.

Example. $\log_2 40 - \log_2 5 = \log_2\tfrac{40}{5} = \log_2 8 = 3$.

For nonnegative reals, $\dfrac{a_1+\cdots+a_n}{n}\ge\sqrt[n]{a_1\cdots a_n}$, equality when all equal.

In plain terms. The average is always at least the geometric mean, so "fixed sum" or "fixed product" extremes happen when the numbers are equal.

Example. For $x>0$, $x + \tfrac9x \ge 2\sqrt{x\cdot\tfrac9x} = 6$, with the minimum $6$ at $x=3$.