Eleventh Grade · Math · week 28

Properties of logarithms

Apply the product, quotient, and power properties of logarithms to expand and condense expressions.

Logarithms turn multiplication into addition, division into subtraction, and exponents into multiplication out front -- three properties that mirror the exponent rules you already know, because logarithms and exponents are two sides of the same coin. The product property: log_b(MN) = log_b(M) + log_b(N). The quotient property: log_b(M/N) = log_b(M) - log_b(N). The power property: log_b(M^p) = p*log_b(M).

"Expanding" means breaking one complicated logarithm into several simpler ones using these properties; "condensing" runs the exact same properties in reverse, combining several logarithms into one. Both directions use the identical three rules -- you are just choosing which way to read them.

📋 For the teacher
Teach it (10–15 min):
  • Introduce all three properties side by side in a small table (product/sum, quotient/difference, power/multiply-out-front) so the 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 to themselves 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 a logarithm of a SUM, only of a product, quotient, or power.
  • Connect back to the exponent rules from months ago explicitly: "multiplying powers adds exponents" is the same underlying structure as "the log of a product is a sum of logs."
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 is there no property for the logarithm of a sum?

Mastery looks like: They correctly apply the product, quotient, and power properties to expand or condense logarithmic expressions in either direction.

Common stumbles: Trying to split log_b(M+N) into separate logs (no such property exists); forgetting to apply the power property to a fractional 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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