Log Calculator

How the log calculator works

A logarithm answers the question “what power do I raise the base to, to get this number?” log₁₀ 1000 = 3 because 10³ = 1000. The calculator solves the equation log_b(x) = y for any one of its three parts:

• Logarithm y: give the number x and pick a base: 10 (the common log), e (the natural log, ln), 2 (the binary log) or any other positive base except 1.
• Number x (the antilog): give the base and the exponent y, and it works out x = b^y. Very large or very small answers, such as 10⁵⁰⁰, are shown in scientific notation.
• Base b: give x and y, and it finds the base with b^y = x.

The result also lists ln x, log₁₀ x and log₂ x together, and the working shows the change of base with your numbers. For the inverse operation, roots, see the root calculator; for logs inside a longer expression, use the scientific calculator.

The definition and the change-of-base formula

b

the base: positive and not 1

x

the number: positive

y

the logarithm (the exponent): any real number

Calculators have keys only for ln and log₁₀. Any other base comes from the change-of-base formula, which works with logs of any one base as long as the same one is used above and below the line:

Solving for the other two parts follows from the definition: the number is x = b^y, and the base is b = x^(1 ÷ y), which can also be found from ln b = ln x ÷ y.

Why x must be positive and b cannot be 1. A positive base raised to any real power is positive, so no power of b is zero or negative: log(0) and logs of negative numbers are not defined in the real numbers. And every power of 1 is 1, so base 1 can never reach any other number.

Exact answers. In floating-point arithmetic ln 1000 ÷ ln 10 comes out as 2.9999999999999996 and ln 125 ÷ ln 5 as 3.0000000000000004. The calculator reads the numbers exactly as typed and checks candidate answers with whole-number arithmetic (5³ really is 125), so it reports 3, not a number a hair away from it. Fractions are found too: log₄ 8 = 3/2 because 4³ = 8², and log₈ 2 = 1/3.

Worked example: log₁₀ 1000

1. 1000 = 10 × 10 × 10 = 10³, so log₁₀ 1000 = 3.
2. The natural log of the same number: ln 1000 ≈ 6.907755279.
3. Change of base: ln 1000 ÷ ln 10 ≈ 6.907755279 ÷ 2.302585093 = 3.
4. Its binary log: log₂ 1000 = 3 ÷ log₁₀ 2 ≈ 9.965784285, just under 10 because 2¹⁰ = 1024.

Other bases. log₂ 1024 = 10 and log₁₀ 0.001 = −3 exactly (10⁻³ = 1/1000). For a base with no key, log₃ 243 = ln 243 ÷ ln 3 = 5, because 3⁵ = 243.

Solving for the other parts. The antilog of 3 in base 10 is 10³ = 1000. If log_b 1000 = 3, the base is 1000^(1/3) = 10.

Where logarithms are used

Logarithms turn questions about repeated multiplication into questions about counting. The clearest everyday case is growth: money earning a steady rate r per period is multiplied by (1 + r) each period, so the number of periods needed to double it is:

At 7% per year that is ln 2 ÷ ln 1.07 ≈ 10.24 years. The familiar rule of 72 (72 ÷ 7 ≈ 10.29) is a quick approximation of the same logarithm. The compound interest calculator shows the balance year by year.

In this calculator, choose Other as the base, enter 1.07 and set x to 2 to get the same 10.24. Very large and very small numbers are also easier to handle as powers of ten; see the scientific notation calculator.

Limits

x and a custom base can be up to 10³⁰⁰, and y can be from −1,000,000 to 1,000,000. Only real logarithms are calculated; logs of negative numbers exist only as complex numbers and are not covered. Exact answers are found when the result is a whole number or a fraction with a denominator up to 12 and the powers involved stay below about 1,200 digits; other answers are shown to about 10 digits.