What a logarithm is
The logarithm log_b(x) answers the question: to what power must the base b be raised to produce x? For example, log₁₀(1000) = 3 because 10³ = 1000. Any base can be computed with the change-of-base rule: log_b(x) = ln(x) ÷ ln(b).
The change-of-base rule matters in practice because most calculators only offer natural log and base-10 log. Dividing one natural log by another cancels the choice of base, so ln(1000) ÷ ln(10) returns the same 3 you would get from a dedicated base-10 key. A logarithm simply asks what power the base must be raised to, so it undoes the exponent calculator.
How to use this calculator
Enter the number x (which must be positive) and the base (positive and not equal to 1). Common choices are base 10 (common log), base e ≈ 2.718 (natural log), and base 2. The result is the exponent that recreates your number.
The badge restates the answer as a power so you can verify it at a glance: base raised to the result should return your original number. Only exact powers of the base give whole-number answers; results are rounded to six decimals. Fractional exponents are roots, which is why the nth root calculator is the other side of the same coin.
Logarithms and exponents
A logarithm is the inverse of an exponent. If b^y = x, then log_b(x) = y. This makes logs useful for turning multiplication into addition and for handling numbers across huge ranges, such as the values shown by the scientific notation calculator.
That compression is why logarithmic scales appear wherever quantities span many orders of magnitude — pH, decibels, the Richter scale. Each step up multiplies by the base rather than adding.
Frequently asked questions
- What is log base 10 of 1000?
- 3, because 10³ = 1000.
- How does the change-of-base formula work?
- log_b(x) = ln(x) ÷ ln(b). Divide the natural log of the number by the natural log of the base.
- Why must the number be positive?
- Logarithms of zero or negative numbers are undefined in the real numbers.
- Why can't the base be 1?
- 1 raised to any power is always 1, so it can never produce other numbers — a base of 1 is undefined for logs.
- What does a negative logarithm mean?
- It means the number is smaller than 1, so the base has to be raised to a negative power to reach it. For example log₁₀(0.001) = −3, because 10⁻³ = 0.001. The result is still perfectly valid — only the input must stay positive.