Logarithm
Functions & Advanced AlgebraA 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 $x$?"
Example
$\log_2(8) = 3$ because $2^3 = 8$. $\log_{10}(1000) = 3$ because $10^3 = 1000$. $\log_5(25) = 2$ because $5^2 = 25$.
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 to the exponent.
Definition
The logarithm base $b$ of $x$, written $\log_b(x)$, is the exponent $y$ such that $b^y = x$. Defined for $x > 0$, $b > 0$, $b \neq 1$. Key identities: $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$. The logarithm converts multiplication to addition: $\log_b(xy) = \log_b(x) + \log_b(y)$.
Example
Solve $3^x = 81$: take log base $3$ of both sides: $x = \log_3(81) = \log_3(3^4) = 4$. Solve $2^x = 50$: $x = \log_2(50) = \ln(50)/\ln(2) \approx 3.912/0.693 \approx 5.64$.
Key Insight
Logarithms tame exponential growth. The Richter scale (earthquakes), decibel scale (sound), and pH scale (acidity) are all logarithmic, because the physical quantities span many orders of magnitude.
Definition
The logarithm $\log_b: (0,\infty) \to \mathbb{R}$ is defined as the inverse of the exponential $b^x$. Analytically, $\ln(x) = \int_1^x (1/t)\,dt$, 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
The complex logarithm $\text{Ln}(i) = \ln|i| + i\,\text{Arg}(i) = 0 + i\pi/2 = i\pi/2$. This is the principal value; other branches add multiples of $2\pi i$.
Key Insight
The logarithm's integral definition explains why $\log(xy) = \log(x) + \log(y)$: the integral from $1$ to $xy$ splits as integral from $1$ to $x$ plus integral from $1$ to $y$ (by substitution). This reveals the algebraic property geometrically.