How to use the Log Calculator
Enter the number and the base. Use 10 for common logs, 2 for binary logs or 2.71828 for the natural log, and read the exponent that the base needs.
What a logarithm answers
A logarithm asks: what power do I raise the base to in order to get this number? Since 10 × 10 × 10 = 1,000, log base 10 of 1,000 is 3. Since 2 × 2 × 2 = 8, log base 2 of 8 is also 3. Logs turn multiplication into addition, which is why they were so useful before calculators.
Solving for an unknown exponent
Logs let you solve equations like 2 to the power x equals 50. Take log base 2 of 50 and you get about 5.64. You meet logs in the pH scale, decibels, earthquake magnitudes and compound interest, all of which compress huge ranges into small, readable numbers.
The formula
log_b(x) = ln(x) ÷ ln(b)
Examples
- log base 10 of 1000
log base 10 of 1,000 = 3 log₁₀ = 3 ln = 6.90775528 log₂ = 9.96578428
- log base 2 of 64
log base 2 of 64 = 6 log₁₀ = 1.80617997 ln = 4.15888308 log₂ = 6
- log base 5 of 200
log base 5 of 200 = 3.29202967 log₁₀ = 2.30103 ln = 5.29831737 log₂ = 7.64385619
Limitations and assumptions
Computed in your browser with standard double-precision arithmetic unless the page says otherwise. Very large or very small values may be rounded.
Frequently asked questions
What is a logarithm?
The inverse of exponentiation: if b^y = x, then log base b of x is y.
How do I change the base?
log_b(x) = ln(x) ÷ ln(b), which is the formula the tool uses.
Why is the log of zero or a negative number invalid?
No real power of a positive base produces zero or a negative result.
What are common bases?
Base 10 for the common log, base e for the natural log and base 2 in computing.