Logarithms explained: laws, examples and solving equations
Understand logarithms as the inverse of powers, use the log laws and solve simple exponential equations. Algebra 2 and A-Level, with examples.
High SchoolA-LevelAPCurriculum fit: Not part of GCSE Maths. US: Algebra II (Building Functions, exponential and logarithmic functions) and AP Precalculus. UK: A-Level Maths.
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What is it?
A logarithm answers the question: what power do I raise the base to in order to get this number? If 2³ = 8, then log base 2 of 8 is 3, written log₂ 8 = 3.
Logarithms are the inverse of exponentials. They let you solve equations where the unknown is in the power, and they are used in science for scales such as pH and decibels.
Key formulas and ideas
- log_b x = y means bʸ = x.
- log a + log b = log(ab)
- log a − log b = log(a/b)
- n log a = log(aⁿ)
- log_b 1 = 0 and log_b b = 1.
Worked examples
Example 1: Evaluate
Question: Work out log₂ 32.
- Ask: 2 to what power gives 32?
- 2⁵ = 32.
Answer: 5
Example 2: Using the laws
Question: Simplify log 4 + log 25 (base 10).
- Add logs by multiplying: log(4 × 25) = log 100.
- 10² = 100, so log 100 = 2.
Answer: 2
Example 3: Solving an exponential equation
Question: Solve 2ˣ = 20 (to 2 d.p.).
- Take logs of both sides: x log 2 = log 20.
- x = log 20 ÷ log 2 = 1.3010 ÷ 0.3010.
- x ≈ 4.32.
Answer: x ≈ 4.32
Common mistakes
- Writing log(a + b) as log a + log b.
- Taking the logarithm of zero or a negative number, which is not possible.
- Mixing up the base and the number.
- Forgetting to use brackets when typing log expressions into a calculator.
Practice questions
Frequently asked questions
What is the difference between log and ln?
log usually means base 10 and ln means the natural logarithm, with base e (about 2.718). Check the convention your course uses.
Why are logarithms useful?
They turn multiplication into addition and powers into multiplication, which helps with very large or small ranges and with solving for exponents.
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Reviewed by StudyMorrow editorial review on 5 Nov 2026.