Logarithm
A logarithm answers the question "to what power must the base be raised to get this number?" and is written as $\log_b(x) = y$ meaning $b^y = x$.
Formula
\log_b(x) = y \text{ means } b^y = x
Definition
A logarithm tells you what exponent is needed: $\log_b(x)$ asks, "what power do I raise $b$ to in order to get $x$?" Formally, the logarithm base $b$ of $x$, written $\log_b(x)$, is the exponent $y$ such that $b^y = x$, defined for $x > 0$, $b > 0$, $b \neq 1$, with key identities $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$, and it converts multiplication to addition: $\log_b(xy) = \log_b(x) + \log_b(y)$. Analytically, $\log_b: (0,\infty) \to \mathbb{R}$ is the inverse of $b^x$, with $\ln(x) = \int_1^x (1/t)\,dt$ and $\log_b(x) = \ln(x)/\ln(b)$; in complex analysis, the complex logarithm $\text{Ln}(z) = \ln|z| + i\,\text{Arg}(z)$ is the principal branch of the multivalued function $\log(z)$.
Example
$\log_2(8) = 3$ because $2^3 = 8$, $\log_{10}(1000) = 3$ because $10^3 = 1000$, and $\log_5(25) = 2$ because $5^2 = 25$. To solve $3^x = 81$, take $\log_3$ of both sides: $x = \log_3(81) = \log_3(3^4) = 4$; to solve $2^x = 50$, $x = \log_2(50) = \ln(50)/\ln(2) \approx 5.64$. The complex logarithm $\text{Ln}(i) = \ln|i| + i\,\text{Arg}(i) = i\pi/2$ is the principal value, with other branches adding multiples of $2\pi i$.
Key Insight
Logarithm and exponentiation are opposite operations, like addition and subtraction: if the exponent builds up a number, the logarithm breaks it back down. Logarithms tame exponential growth, which is why the Richter scale, decibel scale, and pH scale are all logarithmic, since the physical quantities they measure span many orders of magnitude. The logarithm's integral definition explains why $\log(xy) = \log(x) + \log(y)$ geometrically: the integral from $1$ to $xy$ splits as the integral from $1$ to $x$ plus the integral from $1$ to $y$, by substitution.