Base
In a power expression, the base is the number that is multiplied by itself the number of times indicated by the exponent.
Formula
\text{base}^{\text{exponent}} = \text{power}
Definition
The base is the number being multiplied in a power expression, the number on the bottom, while the exponent tells you how many times to multiply it by itself. In $b^n$, $b$ is the base and $n$ is the exponent, and the base can be any real number (positive, negative, zero, or fractional); our base-$10$ number system is named for the base of its positional notation. In the general theory, the base $b$ of the exponential $b^r$ for real $r$ requires $b > 0$ to be well-defined in $\mathbb{R}$, since $b^{1/2} = \sqrt{b}$ requires $b \ge 0$ and negative bases make $b^r$ complex for non-integer $r$; in complex analysis, $z^w = e^{w \operatorname{Log} z}$, where $\operatorname{Log}$ is the multi-valued principal logarithm.
Example
In $5^3$, the base is $5$: you multiply $5 \times 5 \times 5 = 125$. $(-2)^3 = -8$ (negative base, odd exponent gives a negative result), while $(-2)^4 = 16$ (negative base, even exponent gives a positive result), because pairs of negatives cancel. The choice of base in logarithms determines the unit: base $2$ gives bits, base $10$ gives bels, base $e$ gives nats, and the change-of-base formula $\log_b(x) = \ln(x)/\ln(b)$ converts between any two.
Key Insight
The base is what you start with; the exponent controls how many copies you multiply together. The sign rule for negative bases (even exponent gives positive, odd exponent gives negative) follows directly from pairs of negatives canceling. Base $e$ is "natural" because it is the unique base for which the derivative of $b^x$ at $x=0$ equals $1$, making it the fixed point of calculus's fundamental operation.