Logarithmic Function
Functions & Advanced AlgebraA logarithmic function is the inverse of an exponential function, written as f(x) = log_b(x), and tells you what exponent is needed to produce x from base b.
Formula
f(x) = \log_b(x)
Definition
A logarithmic function is the reverse (inverse) of an exponential function. $\log_b(x)$ asks the question: "$b$ to what power gives $x$?"
Example
$\log_2(8) = 3$ because $2^3 = 8$. $\log_{10}(1000) = 3$ because $10^3 = 1000$. The log "undoes" the exponent.
Key Insight
Logarithms turn multiplication into addition and huge numbers into manageable ones. The Richter scale and decibels (sound) are both logarithmic scales.
Definition
The logarithmic function $f(x) = \log_b(x)$ is defined for $x > 0$, with base $b > 0$, $b \neq 1$. It is the inverse of the exponential $b^x$. Domain: $(0, \infty)$. Range: all reals. The graph passes through $(1, 0)$ and $(b, 1)$.
Example
$f(x) = \log_2(x)$: $f(1) = 0$, $f(2) = 1$, $f(4) = 2$, $f(8) = 3$, $f(1/2) = -1$. The function increases slowly (compared to exponentials). Vertical asymptote at $x = 0$.
Key Insight
Logarithmic and exponential functions are mirror images across $y = x$. If $(a, b)$ is on the exponential graph, $(b, a)$ is on the logarithmic graph.
Definition
The natural logarithm $\ln: (0, \infty) \to \mathbb{R}$ is defined as $\ln(x) = \int_1^x (1/t)\,dt$. It is the inverse of $e^x$. For arbitrary base: $\log_b(x) = \ln(x)/\ln(b)$. Analytically extended to $\mathbb{C}$ as $\ln(z) = \ln|z| + i\arg(z)$, it is multivalued.
Example
Complex logarithm: $\ln(-1) = i\pi$ (principal value), since $e^{i\pi} = -1$. This multivaluedness has consequences in complex analysis, requiring branch cuts.
Key Insight
The logarithm is the key example of a function whose domain excludes $0$ in a deep way: its integral definition reveals why $\log(xy) = \log(x) + \log(y)$ geometrically, as areas under $1/t$.