What Is a Logarithm? How to Evaluate Logs, Step by Step

A logarithm asks: what power do I raise the base to in order to get this number? log₂ 32 asks "2 to what power is 32?" It is the inverse of an exponent.

Try it with your own problem ↓

The steps

Read logᵦ(x) = y as "b to the power y equals x". Rewrite it as bʸ = x.
Ask: what exponent on the base gives the number? List powers of the base if it helps.
That exponent is the answer.
If the number isn't a neat power, use the change of base formula: logᵦ(x) = log(x) ÷ log(b) on a calculator.

A worked example

Evaluate log₂ 32.

This asks: 2 to what power is 32?
List powers of 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32.
2⁵ = 32, so the exponent is 5.

log₂ 32 = 5. Check: 2⁵ = 32. ✓

Why it works

Exponents answer "what do I get when I raise 2 to the 5th power?" Logarithms ask it backward: "what power of 2 gives 32?" They are used for earthquake and sound scales, pH, and for solving for an exponent, such as how long money takes to double.

Try it with your own problem

Open the free The Log Ladder with this example already filled in. It shows every step, and you can type over it with your own numbers.

Work it out: log_2(32)Opens The Log Ladder with this example

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Common mistakes

Mixing up the base and the answer. log₂ 32 = 5, not log₅ 32 = 2.
Taking the log of zero or a negative number. It is undefined: no power of a positive base gives 0 or a negative.
Splitting a log of a sum. log(a + b) is not log a + log b. The rule is log(ab) = log a + log b.
Reading a log as multiplication. log₂ 32 is not log × 32.

Questions people ask

What are the main log rules?

log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n · log a.

What is the difference between log and ln?

log usually means base 10 and ln is the natural log, base e (about 2.718).

What is log₇ 50 using change of base?

log 50 ÷ log 7 ≈ 1.699 ÷ 0.845 ≈ 2.01. This makes sense: 7² = 49, close to 50.

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