The Quotient Rule

Part of the Exponentials, Logs & Sequences section of Coddy's Algebra 2 journey. Lesson 22 of 49.

Take log_2(64/4). Since 64 = 2^6 and 4 = 2^2, the quotient is 2^4: exponents subtract when powers divide. So the log is 4, which is 6 minus 2, the log of the top minus the log of the bottom. That is the quotient rule:

log_b(M/N) = log_b(M) - log_b(N)

It holds for positive M and N, and the reason is the same as for the product rule. If M is b to the power p and N is b to the power q, then M over N is b to the power p minus q. The order matters: the top's log comes first, the bottom's log is taken away.

With a positive letter on top: log_2(x/16) becomes log_2(x) - log_2(16), and log_2(16) is 4, so it is log_2(x) - 4. Do not divide the logs: the log of a quotient is a difference of logs, not a quotient of logs.

Challenge

Rewrite log_3(x/9) with the quotient rule. Type each line in the box under the problem: first the difference of two logs, then with the number's log worked out. You are done when it reads log_3(x) minus a whole number.

Try it yourself

log_3(x/9)

Simplify the expression