Logarithm Calculator

Most calculators give you log base 10 and ln (natural log) but make you compute log base 2 or log base 7 by hand using the change-of-base formula.

Enter any positive number and any positive base (other than 1), and the calculator returns the logarithm. It also shows the change-of-base derivation (log_b(x) = ln(x) / ln(b)) so you can see how it got there. Common bases are quick-selectable: 2 for information theory, e for calculus, 10 for engineering and chemistry.

The inverse direction works too — give a base and a logarithm value and get back the original number (which is just exponentiation). Negative numbers and bases of 0, 1, or negative produce errors with explanations of why those cases are undefined or non-real, since that's typically where people get stuck.

Logarithms are the inverse of exponentiation: log_b(x) is the power to which b must be raised to get x. log_2(8) = 3 because 2^3 = 8. log_10(1000) = 3 because 10^3 = 1000. The natural log (ln) uses base e ≈ 2.71828. The common log (log without subscript, in scientific contexts) usually means base 10; in pure math, it often means natural log. Context determines which — engineers default to base 10, mathematicians to base e. The calculator lets you specify any base to avoid ambiguity.

The change-of-base formula is the trick that lets one calculator handle any base: log_b(x) = ln(x) / ln(b). Most physical calculators only have ln and log₁₀ buttons, so to compute log_2(64) you compute ln(64) / ln(2) = 4.158 / 0.693 = 6. The calculator does this internally and shows the work for educational purposes — if you're learning logs, watching the change-of-base derivation makes the concept click better than just seeing the answer.

The pain this addresses: scientific calculators that hide log_2 behind multiple button presses. Engineering and CS work uses base 2 constantly — entropy in information theory is measured in bits, algorithm complexity is often O(log₂ n), digital signal processing uses powers of 2. Going through change-of-base by hand every time is friction. Direct base-2 entry is quick. Same logic for unusual bases (base 7, base 16) that come up in specific problems and never on a calculator's keypad.

Worked example: a sound at 80 dB is how many times louder than one at 60 dB? Decibels are logarithmic: dB = 10 × log₁₀(intensity ratio). 80 − 60 = 20 dB. 20 = 10 × log₁₀(ratio), so log₁₀(ratio) = 2, ratio = 10² = 100. The 80 dB sound is 100× more intense, not 33% louder. This is why decibel scales matter — they compress huge intensity ranges into manageable numbers, but they require log thinking to interpret. The calculator helps with the underlying math; understanding the scale is the harder part.