About the log calculator
A logarithm answers the question: what power must the base be raised to in order to get a given number? This log calculator finds logarithms to base 10, base e (natural log), base 2 or any base you choose, and can also solve for the number or the base when the other two values are known.
It shows the natural, common and binary logarithm together so you can compare them.
Who should use the log calculator
- Students in algebra, pre-calculus and science.
- Programmers estimating how many bits or steps something needs.
- Anyone working with decibels, pH, earthquake scales or growth rates.
How to use the log calculator: what each input means
- Solve for
- The logarithm, the number (antilog), or the base.
- Base
- 10, e, 2 or a custom positive base other than 1.
- Number x
- Must be greater than zero when you take its logarithm.
- Logarithm y
- The exponent, used when solving for the number or the base.
How the log calculator works: method and formula
log_b(x) = y means b^y = x. Calculators compute any base using the change-of-base rule: log_b(x) = ln(x) ÷ ln(b). Solving for the number simply raises the base to the power y, and solving for the base takes x to the power 1 ÷ y.
- Definition
- log_b(x) = y ⇔ b^y = x
- Change of base
- log_b(x) = ln(x) ÷ ln(b)
- Product rule
- log(xy) = log x + log y
How to read your log calculator result
Logarithms turn multiplication into addition, which is why they are used for scales that span many orders of magnitude. The logarithm of 1 is 0 for any base, and the logarithm of a number between 0 and 1 is negative.
Log Calculator worked example
Find log₁₀(1000), then log₂(8).
- 10^3 = 1000, so log₁₀(1000) = 3.
- Using change of base: ln(1000) ÷ ln(10) = 6.9077553 ÷ 2.3025851 = 3.
- 2^3 = 8, so log₂(8) = 3. In the same way, the natural log of 1000 is 6.9077553.
log₁₀(1000) = 3.
Practical uses of the log calculator
- Computing how many doublings reach a target.
- Converting between pH, decibels and linear values.
- Solving exponential equations such as 2^x = 50.
Assumptions and limitations of this calculator
- Real logarithms only: the number must be positive and the base must be positive and not equal to 1.
- Results are floating-point approximations shown to 12 significant digits.