Logarithm Calculator
Any-base logarithms by change of base — the exponent question run backwards, the three laws printed with numbers, and the inverse verified beside every answer.
Logarithm Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Real logarithms only — no complex arguments
- No logarithmic-equation solving (log_b(x) = c forms)
In short: log₂ 8 = 3 because 2³ = 8. The default case: log₁₀ 100 = 2 exactly. Any base comes from the change-of-base law — log_b x = ln x ÷ ln b — so log₅ 125 = ln 125 ÷ ln 5 = 3. Anchors: log_b 1 = 0 for every base, log_b b = 1, and b^(log_b x) = x, which this page verifies numerically on every run.
Formula
log_b x = y means b^y = x
log_b x = ln x ÷ ln b
b^(log_b x) = x
a logarithm is an exponent with the base kept as context.
Worked Example
- Read the question. log_b x asks: to what power must b be raised to reach x?
- Change the base. any log reduces to the engine’s natural log: divide ln x by ln b — the law is printed with your numbers.
- Verify the inverse. raise b to the answer — you must land back on x, and the check card proves it.
log₂ readings are the information theorist’s currency: a 256-option choice carries log₂ 256 = 8 bits — eight yes/no questions, by the definition.
Strengths & Limits Of This Model
Where this engine is strong
- Change-of-base printed with the entered values
- Inverse verified numerically beside every answer
Where it stops
- No expression arguments (log of a product must be computed piecewise)
- Single base per run
Practical Use Cases
Decibels and pH
both are base-10 logs of a ratio
Information
bits are log₂ of the number of choices
Algorithm cost
binary search is O(log n) for a reason
Methodology & Editorial Standards
Refuse a non-positive argument and a base that is non-positive or 1. Compute ln x ÷ ln b; print the change-of-base division with the actual values; detect exact integer answers by rounding within 1e-12 and verify the inverse b^(log_b x) numerically on every run.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Logarithm Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What is a logarithm, in one sentence?
The exponent question run backwards: log_b x is the power you must raise b to in order to get x. log₂ 8 = 3 because 2³ = 8 — same fact, read in the other direction.
Why is the log of 1 always zero?
Because b⁰ = 1 for every valid base: zero is the power that lands on 1. Change the base, the answer never moves: log₂ 1 = 0, log₁₀ 1 = 0, ln 1 = 0.
Why are zero and negative arguments refused?
No positive base raised to any real power lands at zero or below — the graph never crosses. The limit as x slides to 0+ is minus infinity, which is why the refusal says ‘above zero’ and means it.
Why is base 1 refused?
Because 1 to any power is 1: log₁ would have to answer ‘what power of 1 makes x’ with either nothing (x ≠ 1) or everything (x = 1). It answers no question, so it is not a base. Base 0 and negative bases fail the same way.
What is change of base, and why does it work?
log_b x = ln x ÷ ln b — any logarithm reduces to any other base’s. It works because a logarithm IS an exponent: if b^y = x then taking ln of both sides gives y · ln b = ln x, and solve for y. Your calculator only needs one log button; this page prints the division with your numbers.
What are the three log laws?
Product: log(mn) = log m + log n. Quotient: log(m/n) = log m − log n. Power: log(mᵏ) = k · log m. They are the exponent laws read backwards — logs turn multiplication into addition, which is why slide rules worked and why logs linearize growth. Sample: log₁₀ 2 + log₁₀ 5 = log₁₀ 10 = 1, exactly.
Logarithm page or Natural Log page?
This page answers the any-base question and carries the laws; the Natural Log page owns base e specifically — the constant e, the e^(ln x) inverse pair, and the growth-and-decay reading. They share the refusals and the inverse idea; they answer different questions.
Where do logarithms actually appear?
Everywhere scales compress huge ranges: decibels, pH, Richter magnitudes, and musical pitch are all logarithms. Half-life problems invert them (how long until half is left?). And information is defined by them — a guess among n equally likely options carries log₂ n bits.