Logarithm Calculator
A logarithm answers the question “what power do I raise the base to in order to get this number?” Since 10² = 100, log₁₀(100) = 2. This calculator finds a logarithm to any positive base other than 1 and shows the common log, natural log and binary log side by side.
How to use the logarithm calculator
- Enter the number x, which must be greater than 0.
- Enter the base b (10 for common logs, 2 for binary logs).
- Read log_b(x), then check the steps to see the change-of-base formula and a verification.
Formula
logb(x) = ln(x) ÷ ln(b). Key rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(xn) = n·log x; logb(b) = 1; logb(1) = 0.
Worked examples
Base 2
log₂(8) = 3 because 2³ = 8. The tool checks this with 2^3 ≈ 8.
Common log
log₁₀(1000) = 3, and the pH scale uses this: pH = −log₁₀[H⁺].
Understanding logarithms
Logarithms compress huge ranges into manageable numbers. On a base-10 scale, 10, 100, 1,000 and 1,000,000 become 1, 2, 3 and 6. This is why they measure quantities that span many orders of magnitude.
Real-world scales
| Scale | Formula | Meaning |
|---|---|---|
| pH | −log₁₀[H⁺] | Each step is ten times more acidic |
| Decibels | 10·log₁₀(I/I₀) | +10 dB is ten times the intensity |
| Richter | log₁₀(amplitude) | Each whole number is ten times stronger |
Solving exponential equations
To solve 2ˣ = 50, take the logarithm of both sides: x = log(50) ÷ log(2) ≈ 5.644. Logarithms turn exponents into multiplication, which is why they were used for calculation before calculators existed.
Key values to remember
log₁₀(2) ≈ 0.301, log₁₀(3) ≈ 0.477 and ln(2) ≈ 0.693. Using log rules, you can estimate log₁₀(6) = 0.301 + 0.477 = 0.778 without a calculator.
Compound growth
To find how long money takes to reach a target, solve (1 + r)ᵗ = target ratio for t: t = ln(ratio) ÷ ln(1 + r). At 7% a year, doubling takes ln 2 ÷ ln 1.07 ≈ 10.2 years, the exact version of the Rule of 72.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Find log₂(64).
Show answer
log base 2 of 64: 6
- Find log₅(125).
Show answer
log base 5 of 125: 3
Common mistakes to avoid
- Trying to take the log of zero or a negative number, which is undefined for real numbers.
- Using base 1, which cannot work because 1 to any power is 1.
- Confusing log (often base 10) with ln (base e). Check which your course uses.
Frequently asked questions
What is a logarithm?
The inverse of exponentiation: if b^y = x then log_b(x) = y.
What is the difference between log and ln?
log usually means base 10 and ln is the natural logarithm, base e ≈ 2.71828.
Where are logarithms used?
Earthquake magnitude, sound decibels, pH, compound growth and information theory.
Why is log(0) undefined?
No power of a positive base equals 0, so there is no answer.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.