What is a Logarithm?
A logarithm answers the question: to what power must a base be raised to get a given number? Logarithms are the inverse of exponentiation and appear in growth, decay, pH, sound intensity (decibels) and computer science (binary search).
The logarithm log_b(x) = y means b^y = x — the exponent y that the base b must be raised to in order to produce x. For example log_2(8) = 3 because 2³ = 8.
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Step-by-step worked examples
Find log_2(8).
log_2(8) = y means 2^y = 8 2³ = 8, so y = 3
Find log_10(1000).
log_10(1000) = y means 10^y = 1000 10³ = 1000, so y = 3
Solve for x: log_3(x) = 4.
log_3(x) = 4 means 3⁴ = x 3⁴ = 81, so x = 81
Flashcards
Quick quiz
Q1.log_2(16) equals:
Q2.log_10(100) equals:
Q3.log_b(1) is always:
Q4.log_b(mn) equals:
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Common mistakes
Thinking log(mn) = log(m) × log(n). — Correct: The product rule adds logs: log(mn) = log(m) + log(n).
Taking the log of a negative number or zero. — Correct: Logarithms are only defined for positive arguments (x > 0).
Confusing log base 10 with natural log (ln). — Correct: log (no subscript) usually means base 10; ln always means base e.
Forgetting the base changes the answer. — Correct: log_2(8)=3 but log_10(8)≈0.9 — always check which base is being used.
FAQ
What is a logarithm?
The inverse of exponentiation: log_b(x) = y means b raised to y equals x.
What is the logarithm formula?
log_b(x) = y is equivalent to b^y = x.
How do you calculate a logarithm?
Use the change-of-base formula: log_b(x) = ln(x)/ln(b), or recognize b^y = x directly.
What are examples of logarithms?
log_2(8)=3, log_10(1000)=3, and log_3(81)=4 are common worked examples.