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What is High School / Foundations

a Logarithm?

The inverse of exponentiation — asking "what power do I need to raise this base to in order to get that number?"

Simple Definition

You already know that \(2^3 = 8\). A logarithm asks the reverse question: "2 to the power of what gives 8?" The answer is 3, and we write it as \(\log_2 8 = 3\).

In general, \(\log_b x = y\) means \(b^y = x\). The logarithm is the exponent. It's nothing more exotic than that — just exponentiation turned around.

Logarithms are indispensable in science because they tame enormous numbers. The Richter scale, decibels, pH, and stellar magnitude are all logarithmic scales — because the world spans many orders of magnitude, and logarithms compress that range into something manageable.

The Core Equivalence

\(\log_b x = y\)

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\(b^y = x\)

These two statements say exactly the same thing. Switching between them is the most important skill in logarithm problems. If you're stuck, convert to exponential form.

Intuition

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Counting the zeros

\(\log_{10}(1{,}000{,}000) = 6\) — because \(10^6 = 1{,}000{,}000\). The log base 10 of a number roughly counts the number of digits minus one. \(\log_{10}(1000) = 3\), \(\log_{10}(10{,}000) = 4\). Logs convert multiplication into addition — that's why slide rules worked.

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Inverse functions

\(e^x\) and \(\ln x\) are inverses: \(e^{\ln x} = x\) and \(\ln(e^x) = x\). They undo each other completely. On a graph, \(y = \ln x\) is the reflection of \(y = e^x\) across the line \(y = x\). This mirror relationship is at the heart of why logarithms appear in integration.

Three Important Types

Common Log

\(\log_{10} x\)

Written as \(\log x\). Used in pH, Richter scale, decibels. Base 10.

Natural Log

\(\ln x\)

Base \(e \approx 2.718\). The most important log in calculus. Derivative: \(\frac{1}{x}\).

Binary Log

\(\log_2 x\)

Base 2. Used in computer science and information theory. \(\log_2 1024 = 10\).

Log Rules

y = ln(x) grows without bound — but very slowly. Domain: x > 0 only.

Product	\(\log_b(xy) = \log_b x + \log_b y\)
Quotient	\(\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y\)
Power	\(\log_b(x^n) = n\log_b x\)
Change of Base	\(\log_b x = \dfrac{\ln x}{\ln b} = \dfrac{\log x}{\log b}\)
Inverse	\(b^{\log_b x} = x \quad\text{and}\quad \log_b(b^x) = x\)
Special values	\(\log_b 1 = 0 \quad\text{and}\quad \log_b b = 1\)

Quick Examples

Example 1 — Basic

\(\log_3 81\)

\(3^? = 81 = 3^4\)
Answer: \(4\)

Example 2 — Product rule

\(\log_2 4 + \log_2 8\)

\(= \log_2(4 \cdot 8) = \log_2 32 = 5\)

Example 3 — Solving

\(\ln x = 3\)

Exponentiate both sides:
\(x = e^3 \approx 20.09\)

Example 4 — Derivative

\(\dfrac{d}{dx}[\ln(x^2+1)]\)

Chain rule: \(\dfrac{1}{x^2+1} \cdot 2x = \dfrac{2x}{x^2+1}\)

Common Mistakes

✗ log(a + b) = log(a) + log(b)

Completely wrong. The product rule says \(\log(ab) = \log a + \log b\) — multiplication inside becomes addition outside. There is no rule that simplifies \(\log(a+b)\).

✗ log(a) · log(b) = log(ab)

Also wrong. \(\log(ab) = \log a + \log b\), not \(\log a \cdot \log b\). Don't mix up addition and multiplication on either side of this rule.

✗ Forgetting the domain restriction

\(\log x\) is only defined for \(x > 0\). You cannot take the log of zero or a negative number (in the reals). Always check that the argument is positive before evaluating.

Related Concepts

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