Logarithms: The Inverse of Exponential Functions
Logarithms are just exponential functions flipped around. If 2ยณ = 8, then logโ 8 = 3. They answer the question: "What power do I need to get this result?" Common logarithms use base 10, while natural logarithms use base e โ 2.718.
The laws of logarithms are your best friends for simplifying complex expressions. log(UV) = log U + log V means multiplication inside becomes addition outside. Division becomes subtraction, and powers become multiplication in front.
When solving logarithmic equations, remember that logarithms and exponentials cancel each other out. If logโ x = 3, then x = 2ยณ = 8. Always check your answers - logarithms are only defined for positive numbers!
Change of base formula log_a U = (log U)/(log a) lets you use your calculator for any base. Most calculators only have log (base 10) and ln (base e) buttons, but you can find any logarithm using this formula.
Calculator Hack: To find logโ
25 on your calculator, compute log 25 รท log 5 = 2!
Practice with compound logarithmic equations by working from the outside in. If you see log(log(log n)) = 0, start with the outermost log and work your way to the center step by step.