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Algebra · Quick Reference
Logarithms Cheat Sheet
In one line: a logarithm answers the question 'what exponent gives me this number?' — the inverse operation of exponentiation, and the key to solving equations where the variable is stuck in an exponent.
Key Ideas
1Logarithm Definition. log_b(x) = y means the same thing as b^y = x — 'the log base b of x is y'.
2Common Log and Natural Log. log(x) usually means log base 10. ln(x) means log base e (the natural logarithm).
3Product Rule. log_b(m*n) = log_b(m) + log_b(n) — logs turn multiplication into addition.
4Quotient Rule. log_b(m/n) = log_b(m) - log_b(n) — logs turn division into subtraction.
5Power Rule. log_b(m^n) = n * log_b(m) — logs turn exponents into multiplication out front.
Worked Examples
Evaluate log_2(8).
log_2(8) = 3
Evaluate log_10(1000).
log_10(1000) = 3
Use the product rule to simplify log_2(4) + log_2(8).
log_2(32) = 5
Formulas
log_b(x) = y means b^y = x
log_b(m*n) = log_b(m) + log_b(n)
log_b(m/n) = log_b(m) - log_b(n)
log_b(m^n) = n * log_b(m)
Practice Yourself
Evaluate log_2(16).
4
Evaluate log_10(100).
2
Evaluate log_3(9).
2