Log Calculator

A logarithm answers the question “what power gives this number?”. log_b(x) = y means bʸ = x. So log₁₀ 1000 = 3 because 10³ = 1000, and log₂ 8 = 3 because 2³ = 8. Logarithms turn multiplication into addition, which is why they appear in scales such as pH, decibels and earthquake magnitude.

Use the change-of-base formula: log_b(x) = ln(x) ÷ ln(b), or equivalently log₁₀(x) ÷ log₁₀(b). For log₃ 50 that is ln 50 ÷ ln 3 = 3.9120 ÷ 1.0986 ≈ 3.561. Choose Other base in the calculator and it shows these steps for you.

“log” usually means the common logarithm, base 10, while “ln” is the natural logarithm, base e ≈ 2.71828. They differ by a constant factor: ln x = log₁₀ x × ln 10 ≈ 2.302585 × log₁₀ x. In some higher-maths and programming contexts “log” means ln, so check the convention you are using.

The antilog reverses a logarithm: if log_b(x) = y, then the antilog is x = bʸ. For base 10, antilog(2.5) = 10^2.5 ≈ 316.23; for the natural log the antilog is eʸ. Antilog mode calculates bʸ for any base, including results far too large or small for an ordinary decimal, which it gives in scientific notation.

Take the logarithm of both sides so the exponent comes down: x = log₂ 32 = 5. In general bˣ = y gives x = log_b(y) = ln y ÷ ln b. For a · b^(kx) = y, divide by a first and then by k: x = log_b(y/a) ÷ k. Solve mode does each step and checks the answer.

For a positive base, bʸ is positive for every real y, so no real exponent can produce zero or a negative number. That means log_b(x) is only defined for x > 0. As x approaches 0 the logarithm heads towards minus infinity. In complex numbers, logs of negatives exist, but this calculator works in the real numbers.

The main rules are: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n · log x, log_b b = 1, log_b 1 = 0, and the change of base log_b x = ln x / ln b. They hold for any valid base, and the Log rules panel lists each one with a worked example.

Base-2 logarithms count doublings, so they appear throughout computing: log₂ n is the number of bits needed to represent n values and the number of steps a binary search takes on n items. log₂ 1024 = 10 because 2¹⁰ = 1024. They are also used for musical octaves and for information measured in bits.

For a common (base-10) logarithm, the characteristic is the integer part and the mantissa is the fractional part. log₁₀ 250 = 2.39794, so the characteristic is 2 (250 has three digits) and the mantissa is 0.39794, which is log₁₀ 2.5. Printed log tables list only mantissas, because the characteristic just shifts the decimal point.

Solve (1 + r)ˣ = 2 for x, giving x = ln 2 ÷ ln(1 + r). At 5% per year that is 0.6931 ÷ 0.04879 ≈ 14.21 years; at 8% it is about 9.01 years. The rule of 72 (72 ÷ the rate) gives a quick estimate. In Solve mode, enter base 1.05 and right side 2.