Logarithm Calculator

What Is a Logarithm?

A logarithm answers the question: to what power must a base be raised to produce a given number? log base a of x asks: a to what power equals x? log₁₀(1000) = 3 because 10³ = 1000. log₁₀(100) = 2 because 10² = 100. ln(e²) = 2 because e² = e squared. The natural logarithm (ln) uses the base e ≈ 2.718. The common logarithm (log or log₁₀) uses base 10. Key rules: log(a×b) = log(a) + log(b). log(a/b) = log(a) − log(b). log(aⁿ) = n × log(a). These rules convert multiplication into addition and are wh

Change of Base Formula

logₐ(x) = log(x) / log(a) = ln(x) / ln(a). Any base can be computed using log₁₀ or ln on a calculator. Example: log₂(32) = log(32)/log(2) = 1.50515/0.30103 = 5.

Logarithm Laws

Product: log(ab) = log(a) + log(b). Quotient: log(a/b) = log(a) − log(b). Power: log(aⁿ) = n·log(a). These rules convert multiplication into addition — the original purpose of logarithm tables before calculators.

Exam Tips and Common Errors

Common mathematical errors to avoid: sign errors when moving terms across an equation (changing sign), order of operations (BIDMAS/BODMAS — brackets, indices, division and multiplication, addition and subtraction), not checking whether answers are reasonable (a negative length or probability outside 0-1 indicates an error), and rounding too early in multi-step calculations (carry extra decimal places until the final step). Always substitute your answer back into the original equation or problem