Expanding Logarithms

Pre-Calculus · Axiom Academy

Use the product, quotient, and power rules to break a single complex logarithm into a sum and difference of simpler logs. Expand completely, writing the expression as a sum and difference of logarithms with no products, quotients, or powers left inside any log: Nicely done. You expanded one nested logarithm into three clean terms by peeling off one rule at a time. Outside in: attack the outermost operation first — here the division — then work inward to products, then powers. Quotient becomes subtraction: , so the denominator's log is subtracted. Exponents become coefficients: the Power Rule drops each exponent out front, turning into and into . The same three rules expand any log of products, quotients, and powers — and run in reverse to condense an expression back into a single logarithm.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.