Logarithms from Zero to Mastery

Logarithms intimidate students because the notation $\log_a b$ doesn't intuitively reveal what's happening. The truth is, logarithms are just exponents in disguise. Once you crack that idea, every log rule follows from familiar exponent rules. This guide builds logs from the ground up.

The definition (memorise this one)

$\log_a b = c \iff a^c = b$

In words: "$\log_a b$ is the exponent you raise $a$ to in order to get $b$." That's it. Everything else is bookkeeping.

Examples:

• $\log_2 8 = 3$ because $2^3 = 8$.
• $\log_{10} 1000 = 3$ because $10^3 = 1000$.
• $\log_5 1 = 0$ because $5^0 = 1$.

Common bases

• $\log$ (no subscript): usually $\log_{10}$ in pre-calculus, but $\log_e = \ln$ in higher math (calculus, physics, ML). Check your textbook's convention.
• $\ln$ (natural log): $\log_e$, where $e \approx 2.71828$. The "natural" base because $\frac{d}{dx}\ln x = \frac{1}{x}$ — clean derivative.
• $\log_2$: computer science (binary), information theory.

The four core rules

All four come from exponent rules ($a^m \cdot a^n = a^{m+n}$, etc.) reversed.

1. Product rule

$\log_a (xy) = \log_a x + \log_a y$

Multiplication inside the log → addition outside. (Mirror of $a^m a^n = a^{m+n}$.)

2. Quotient rule

$\log_a \frac{x}{y} = \log_a x - \log_a y$

Division → subtraction.

3. Power rule

$\log_a (x^n) = n \log_a x$

Exponent comes outside as a multiplier. Most useful for solving log equations.

4. Change of base

$\log_a b = \frac{\log_c b}{\log_c a}$

For any reference base $c$. Lets you compute $\log_7 50$ on a calculator that only has $\log_{10}$ or $\ln$.

Solving log equations

The standard playbook:

If the equation has multiple log terms, condense them into a single log using rules 1–3, then convert to exponential form.

Example: $\log_2(x) + \log_2(x - 2) = 3$.

• Condense: $\log_2 (x(x-2)) = 3$.
• Exponential form: $x(x - 2) = 2^3 = 8$.
• Quadratic: $x^2 - 2x - 8 = 0$, factor: $(x - 4)(x + 2) = 0$, so $x = 4$ or $x = -2$.
• Check domain: $\log_2(-2)$ undefined (logs need positive arg), so reject $x = -2$.
• Answer: $x = 4$.

Always check the domain — squaring or condensing logs can introduce extraneous solutions that violate the positive-argument requirement.

Useful identities

• $\log_a 1 = 0$ (anything to the zero is 1).
• $\log_a a = 1$ (anything to the first is itself).
• $\log_a a^n = n$ (the inverse identity).
• $a^{\log_a x} = x$ (the inverse identity, the other way).

Why logs matter

• Compress huge ranges: pH, decibels, Richter scale, magnitudes — all logarithmic because the underlying quantities span many orders of magnitude.
• Linearise exponential data: log-axis plots reveal exponential trends as straight lines. Standard in finance, biology, machine learning.
• Calculus: $\frac{d}{dx}\ln x = \frac{1}{x}$ — the cleanest derivative on the planet, worth memorising forever.
• Information theory: log base 2 measures bits; log base $e$ measures nats.

Common mistakes

• $\log(x + y) \neq \log x + \log y$. The product rule is for $\log(xy)$, not $\log(x+y)$. There's no "log of a sum" rule.
• Negative arguments: $\log_a(-3)$ is undefined in the reals.
• Forgetting to check domain when solving equations.

Try it yourself

Drop any log expression into our Equation Solver — it picks the right rule chain and walks you through.

Related:

• Glossary: Logarithm
• Glossary: Exponent
• Algebra Formulas Cheat Sheet