Logarithms

When we folded paper, we asked: how many times do you have to fold the paper to get 32 layers? In other words: what power do you have to raise 2 to in order to get 32? The answer is 5, because 2⁵ = 32.

A logarithm answers exactly this kind of question. We write it as log₂ 32 = 5.

logₐ b = c means aᶜ = b.

We read it as "log base a of b equals c".

So a logarithm is an exponent: the number you have to raise the base a to in order to get b.

Contents

How to work out a logarithm

Method: turn the logarithm into a question about a power.

LogarithmQuestionAnswer
log₂ 82 to what power is 8?3, because 2³ = 8
log₃ 813 to what power is 81?4, because 3⁴ = 81
log₁₀ 1,00010 to what power is 1,000?3
log₅ 15 to what power is 1?0, because 5⁰ = 1
log₂ ⅛2 to what power is ⅛?−3, because 2⁻³ = ⅛

💡 The logarithm of a fraction less than 1 (with a base greater than 1) is negative. The fraction ⅛ is 2⁻³.

When a logarithm exists

  • The base a must be positive and not equal to 1, just as for the exponential function.
  • The argument b must be positive. A power with a positive base is always positive, so neither log₂ (−4) nor log₂ 0 exists.

logₐ b makes sense only if a > 0, a ≠ 1 and b > 0.

Key values

For every allowed base a:

  • logₐ 1 = 0, because a⁰ = 1,
  • logₐ a = 1, because a¹ = a,
  • logₐ aᵏ = k: the logarithm "undoes" the power,
  • a^(logₐ b) = b: the power "undoes" the logarithm.

For example, 3^(log₃ 7) = 7 and 2^(3 + log₂ 5) = 2³ · 2^(log₂ 5) = 8 · 5 = 40.

Logarithms with fractional answers

The answer does not have to be a whole number. In that case, write both the base and the argument as powers of the same number.

Worked example 1

log₄ 8 = ?

  1. We are looking for y such that 4ʸ = 8.
  2. 4 = 2², 8 = 2³, so 2^(2y) = 2³.
  3. 2y = 3, so log₄ 8 = 3/2.

Worked example 2

log₂ √2 = ?

A square root is a power with the exponent ½: √2 = 2^½. So log₂ √2 = ½.

Worked example 3

Log base ⅓ of 9 = ?

  1. (⅓)ʸ = 9, so 3^(−y) = 3².
  2. −y = 2, y = −2.

Equations straight from the definition

Unknown argument

log₂ x = 5. From the definition, x = 2⁵ = 32.

log₃ x = −2. From the definition, x = 3⁻² = 1/9.

Unknown base

logₓ 16 = 2. From the definition, x² = 16. The base must be positive, so x = 4 (−4 does not work).

logₓ ⅛ = 3. From the definition, x³ = ⅛, so x = ½.

The common logarithm and the calculator

The logarithm to base 10 is called the common logarithm, and the base is often left out: log 1,000 = log₁₀ 1,000 = 3. On a calculator it is the log button.

You can work out a logarithm that does not come out nicely on a calculator. For a base other than 10, use the formula

logₐ b = log b / log a

For example, log₂ 5 = log 5 / log 2 ≈ 0.699 / 0.301 ≈ 2.32. You will see why the formula works in the article Laws of Logarithms.

Common mistakes

  • Swapping the base and the argument. log₂ 8 asks "2 to what power is 8", not "8 to what power is 2" (that would be ⅓).
  • logₐ 1 = 1. No, logₐ 1 = 0, because a⁰ = 1.
  • The logarithm of a negative number or of zero. It does not exist.
  • A negative base when the base is unknown. From x² = 16, only x = 4 works.

Summary

  • logₐ b = c means aᶜ = b: a logarithm is an exponent.
  • We need a > 0, a ≠ 1, b > 0.
  • logₐ 1 = 0, logₐ a = 1, logₐ aᵏ = k, a^(logₐ b) = b.
  • For a fractional answer, write the base and the argument as powers of the same number.
  • log with no base is the common logarithm (base 10).

Try it yourself

  1. log₃ 27
  2. log₂ (1/16)
  3. log₉ 27
  4. log₅ x = 3
  5. logₓ 49 = 2
Answers:
  1. 3, because 3³ = 27.
  2. −4, because 2⁻⁴ = 1/16.
  3. 3/2, because 9 = 3², 27 = 3³ and 2y = 3.
  4. x = 5³ = 125.
  5. x = 7.

Practice

👉 Next: Laws of Logarithms