The logarithm calculator computes log₁₀, natural log (ln), log₂, or any custom base logarithm. Also computes antilog (inverse log). Enter a value and select the base to get instant results.
Logarithm Calculator
How to Use the Logarithm Calculator
A logarithm answers the question: "to what power must the base be raised to get x?" log₂(8) = 3 because 2³ = 8.
Common Logarithm (log₁₀)
log₁₀(x) tells how many digits x has (approximately). log₁₀(1000) = 3. log₁₀(1,000,000) = 6. The Richter scale uses log₁₀: an earthquake of magnitude 7 is 10× more powerful than magnitude 6. Decibels use 10×log₁₀(power ratio).
Natural Logarithm (ln)
ln(x) = log_e(x) where e ≈ 2.71828. Appears in compound interest, radioactive decay, entropy, and calculus. ln(e) = 1, ln(1) = 0, ln(e²) = 2. Derivative of ln(x) = 1/x.
Change of Base Formula
log_b(x) = log₁₀(x) / log₁₀(b) = ln(x) / ln(b). Calculate log₅(125): ln(125)/ln(5) = 4.828/1.609 = 3. Check: 5³ = 125. ✓
Frequently Asked Questions
What is a logarithm?
log_b(x) = y means b^y = x. The logarithm answers: 'b raised to what power gives x?' log₁₀(1000) = 3 because 10³ = 1000. ln(e²) = 2 because e² = e². log₂(8) = 3 because 2³ = 8. Logarithms convert multiplication to addition: log(ab) = log(a) + log(b).
What is the natural logarithm (ln)?
ln(x) = log_e(x) where e ≈ 2.71828. It appears naturally in calculus, growth/decay, and probability. ln(x) is the area under 1/t from 1 to x. Key values: ln(1)=0, ln(e)=1, ln(e²)=2. Relationship to log₁₀: log₁₀(x) = ln(x)/ln(10) = ln(x)/2.3026.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted.
What is the change of base formula?
log_b(x) = ln(x)/ln(b) = log₁₀(x)/log₁₀(b). This converts any logarithm to natural log or log₁₀ for calculation. Example: log₅(125) = ln(125)/ln(5) = 4.828/1.609 = 3 (since 5³=125).