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Logarithms

A logarithm asks: what power produces this number? It's the inverse of an exponent — pulling the high-up exponent back down to ground level.

Log and exp are mirrors

Read the same fact two ways. The exponent goes up; the log brings it back down.

Exponent form

23=8

↕ same fact ↕

Logarithm form

log2(8)=3

Base2
Exponent3

logb(x) asks: "what power do I raise b to, to get x?" The answer is the exponent on the other side.

Two ways to say the same thing

2³ = 8 and log₂(8) = 3 are the same statement — written from two angles. The first answers "what is the result?"; the second answers "what was the exponent?"

Reading a log

log_b(x) reads as "log base b of x" and asks: what power do I raise b to, to get x?

  • log₁₀(100) = 2 because 10² = 100.
  • log₂(32) = 5 because 2⁵ = 32.
  • log₃(1) = 0 because 3⁰ = 1.

Common bases

log with no base usually means base 10 (the "common" log). ln means base e (the "natural" log, where e ≈ 2.718). They're the same idea, different bases.

The three log laws

  • Product — log(ab) = log(a) + log(b)
  • Quotient — log(a/b) = log(a) − log(b)
  • Power — log(aⁿ) = n · log(a)

These are mirror images of the exponent laws. Multiplying inside a log becomes addition outside; powers come down as factors.

Why logs matter

They turn multiplication into addition and exponents into multiplication — which made hand-calculations on giant numbers tractable before calculators. Today they show up wherever quantities span many orders of magnitude:

  • The Richter scale for earthquakes is logarithmic.
  • The decibel scale for sound is logarithmic.
  • The pH of a solution is a negative log.

Quick check

  1. What is log₂(64)?
  2. What is log₁₀(0.01)?
  3. Simplify log(8) + log(125) (base 10).

Answers: 6, −2, and 3 (since 8 × 125 = 1000).

Quick check

What is log₂(64)?

Quick check

Simplify log(8) + log(125) (base 10).