Exponent and logarithm laws side by side
Every log law is an exponent law wearing a disguise — here they are paired.
Whenever a log rule doesn't stick, translate it back to the matching exponent rule you already trust: adding exponents on multiplication mirrors adding logs on multiplication, and so on down the list.
For a real number a and positive integers m, n:
a^n = \underbrace{a \cdot a \cdots a}_{n \text{ factors}}The laws of exponents:
- Product rule: a^m \cdot a^n = a^{m+n}
- Quotient rule: \dfrac{a^m}{a^n} = a^{m-n} (for a \neq 0)
- Power rule: (a^m)^n = a^{mn}
- Power of a product: (ab)^n = a^n b^n
- Power of a quotient: \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} (for b \neq 0)
To make the quotient rule consistent for all exponents we define, for a \neq 0: a^0 = 1 \qquad\text{and}\qquad a^{-n} = \frac{1}{a^n}. With these definitions, all five laws hold for every integer exponent.
The exponent rules every log law below is secretly copying.
For a > 0 and integers m, n with n > 0: a^{1/n} = \sqrt[n]{a}, \qquad a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}. Read the denominator as the root and the numerator as the power.
Fractional exponents are radicals in disguise.
Valid whenever both sides are defined:
- Product property: \sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}
- Quotient property: \sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}
- Rational-exponent link: \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m} = a^{m/n}
For b > 0, b \neq 1, and x > 0, \log_b x = y \quad \Longleftrightarrow \quad b^y = x. That is, \log_b x is the exponent to which b must be raised to produce x.
The log-exponential equivalence every other log rule is built on.
Formula
Properties of Logarithms
For b > 0, b \neq 1, positive reals M, N, and any real p:
- Product rule: \log_b(MN) = \log_b M + \log_b N
- Quotient rule: \log_b\frac{M}{N} = \log_b M - \log_b N
- Power rule: \log_b(M^p) = p \log_b M
Product, quotient, and power rules — the exponent laws, mirrored.
Formula
Change-of-Base Formula
For any valid bases a, b and x > 0: \log_b x = \frac{\log_a x}{\log_a b}.
Convert any log into one your calculator actually has a button for.
Because \log_b x and b^x are inverse functions: b^{\log_b x} = x \quad (x > 0), \qquad \log_b(b^x) = x \quad (\text{any real } x).
For b > 0, b \neq 1: b^m = b^n \iff m = n, \qquad \log_b M = \log_b N \iff M = N \ (M, N > 0).
The move that lets you drop the log or exponent from both sides of an equation.
Show proof
Both directions of each statement lean on the pair of inverse identities \log_b(b^x) = x and b^{\log_b x} = x.
Given b^m = b^n, apply \log_b to both sides: \log_b(b^m) = \log_b(b^n), and the inverse identity collapses each side to just the exponent, leaving m = n. The converse — equal inputs to b^{(\cdot)} give equal outputs — holds simply because b^{(\cdot)} is a function.
Given \log_b M = \log_b N, apply b^{(\cdot)} to both sides: b^{\log_b M} = b^{\log_b N}, and the other inverse identity collapses each side back to the original argument, leaving M = N. Again the converse holds because \log_b is a function.
The restriction b \neq 1 is exactly what makes this work: for 0 < b < 1 or b > 1, b^x is strictly increasing or strictly decreasing, so it never repeats an output and the inverse identities are valid. At b = 1 that fails completely — 1^m = 1^n = 1 for every m and n, so equal outputs say nothing at all about the exponents, which is precisely why b = 1 is excluded from the theorem.
An exponential function with base b has the form f(x) = a \cdot b^x, \qquad b > 0,\; b \neq 1,\; a \neq 0, where a = f(0) is the initial value. It models growth when b > 1 and decay when 0 < b < 1.