Twelfth Grade · Math · week 14

Properties of logarithms: expanding, condensing, and change of base

Apply the product, quotient, and power properties of logarithms, and use the change-of-base formula to evaluate a logarithm in any base.

Logarithms turn multiplication into addition, division into subtraction, and exponents into multiplication out front: log_b(MN) = log_b(M)+log_b(N), log_b(M/N) = log_b(M)-log_b(N), and log_b(M^p) = p*log_b(M). "Expanding" breaks one complicated logarithm into several simpler ones with these rules; "condensing" runs the exact same rules in reverse.

Most calculators only compute log (base 10) and ln (base e) directly, so evaluating a logarithm in any OTHER base uses the change-of-base formula: log_b(x) = ln(x)/ln(b) (or log(x)/log(b) -- either works, since the ratio comes out the same either way).

📋 For the teacher
Teach it (10–15 min):
  • Introduce all three properties side by side (product/sum, quotient/difference, power/multiply-out-front) so the underlying pattern -- a logarithm turns one operation into a simpler one -- is visible at a glance.
  • Have them expand and then immediately re-condense the same expression, back to where they started, to prove the properties truly reverse each other.
  • Watch for the classic overreach: log_b(M+N) is NOT log_b(M)+log_b(N) -- there is no property for the logarithm of a SUM.
  • Practice change of base as a fixed template -- "new base's natural log on top, old base's natural log on bottom" -- until it needs no thought.
Talk about it:
  • Why does the product property turn multiplication inside a logarithm into addition outside of it?
  • What is the difference between expanding and condensing a logarithmic expression?
  • Why does the change-of-base formula work no matter which base you convert through (natural log or common log)?

Mastery looks like: They correctly apply the product, quotient, and power properties in either direction, and use change of base to evaluate a logarithm in any base.

Common stumbles: Trying to split log_b(M+N) into separate logs; forgetting to apply the power property to a fractional or negative exponent when expanding.

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

LogarithmsThe inverse of exponents — and what a log scale shows.try →MultiplicationEqual groups and arrays make products finally click.try →DivisionShare fairly and build equal groups.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
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