Introduction to Logarithms and How to Calculate Them

A logarithm answers the question “to what power must the base be raised to get this number?”: log_b x = y means b^y = x. In brief: log₂ 8 = 3 because 2³ = 8, log₁₀ 1000 = 3 because 10³ = 1000, and log₅ (1/25) = −2 because 5⁻² = 1/25. The base b must be positive and not equal to 1, and the argument x must be positive.

Logarithms are the inverse operation of exponentiation, in the same way that subtraction undoes addition. They turn multiplication into addition, make huge ranges of numbers manageable (decibels, pH, the Richter scale) and let you solve equations in which the unknown is an exponent. This guide explains the definition, special values, how to calculate logarithms by hand and with a calculator, and gives a table of common values.

What is a logarithm?

For b > 0, b ≠ 1 and x > 0, the equation y = log_b x is equivalent to b^y = x. The number b is the base, x is the argument and y is the logarithm. The two forms say the same thing:

2⁵ = 32 ⟺ log₂ 32 = 5.

The restrictions come from the exponential function y = b^x, which is positive and takes each positive value exactly once. So you cannot take the logarithm of zero or of a negative number in the real numbers, and the base 1 is not allowed because 1^y is always 1. The logarithm is the inverse of the exponential function.

Two bases have special names: log x or log₁₀ x is the common logarithm (base 10), and ln x is the natural logarithm (base e ≈ 2.718). In computer science, log₂ is often written lg or log.

What are the special values of logarithms?

• log_b 1 = 0, because b⁰ = 1.
• log_b b = 1, because b¹ = b.
• log_b (bⁿ) = n, and b^{log_b x} = x for x > 0.
• For x > 1 and b > 1, log_b x is positive. For 0 < x < 1, it is negative.

How do you calculate a logarithm?

By rewriting as a power

1. Set y = log_b x.
2. Rewrite as b^y = x.
3. Express x as a power of b and read the exponent.

Example 1

log₃ 81 = y means 3^y = 81 = 3⁴, so y = 4.

Example 2

log₄ 8: 4^y = 8, so 2^2y = 2³, 2y = 3 and y = 3/2.

Example 3

log_1/2 16: (1/2)^y = 16 gives 2^−y = 2⁴, so y = −4.

Example 4

log_x 49 = 2 means x² = 49, so x = 7 (the base must be positive).

With a calculator

Calculators have keys for log (base 10) and ln. For other bases, use the change-of-base formula log_b x = ln x/ln b. For example, log₂ 10 = ln 10/ln 2 = 2.3026/0.6931 ≈ 3.3219. Check: 2^3.3219 ≈ 10.

Estimating a logarithm

Between which two integers is log₂ 50? Since 2⁵ = 32 and 2⁶ = 64, the value lies between 5 and 6, close to 5.64. For common logarithms, the number of digits of an integer N is ⌊log N⌋ + 1: log 5,000,000 ≈ 6.699, so 5,000,000 has 7 digits.

Worked examples

Example 5: solving a simple equation

log₂ (x + 1) = 4 means x + 1 = 2⁴ = 16, so x = 15. Check: log₂ 16 = 4 ✓.

Example 6: a growth question

How many years until 1,000 dollars at 5% yearly interest becomes 2,000? 1.05^t = 2, so t = log_1.05 2 = ln 2/ln 1.05 ≈ 14.2 years.

Example 7: pH

pH = −log₁₀[H⁺]. For a hydrogen ion concentration of 10⁻⁷ mol/L, pH = 7. For 3.2 × 10⁻⁴, pH = −log(3.2 × 10⁻⁴) = −(0.505 − 4) = 3.49.

Table of common logarithms

Where do logarithms come from?

John Napier published the first table of logarithms in 1614, motivated by the need to simplify the multiplication of large numbers in astronomy. Henry Briggs soon produced tables of base-10 logarithms. For more than 300 years, logarithm tables and slide rules, which add lengths proportional to logarithms, were the standard tools of calculation, until electronic calculators replaced them in the 1970s. The word logarithm is built from Greek words meaning “ratio number”.

Where are logarithms used today?

Logarithmic scales compress large ranges: the loudness of sound in decibels (L = 10 log(I/I₀)), the acidity pH, the magnitude of earthquakes and the brightness of stars. In computer science, the number of steps in a binary search is about log₂ n: a sorted list of one million entries needs at most 20 comparisons. In finance and science, logarithms are used to solve for time and for growth rates.

Which mistakes are common?

Taking the logarithm of a non-positive number

log_b 0 and log_b (−4) are not defined in the real numbers.

Mixing up the base and the argument

In log₂ 8 the base is 2 and the argument is 8. The result is 3, not 1/3.

Treating log as a multiplication

log₁₀ 100 is not “log times 100”. It is the exponent 2.

Assuming log(a + b) = log a + log b

The correct rule is for products: log(ab) = log a + log b.