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 why logarithms were historically so valuable for simplifying complex calculations by hand, before calculators existed.

What's the key thing to understand about 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.

What should I know about 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.

What's the key thing to understand about 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 to check it makes sense — this catches most arithmetic slips and is worth the extra 30 seconds in an exam.