The Product Rule
Part of the Exponentials, Logs & Sequences section of Coddy's Algebra 2 journey. Lesson 21 of 49.
Take log_3(9 × 27). Since 9 = 3^2 and 27 = 3^3, the product is 3^2 × 3^3, which is 3^5: exponents add when powers multiply. So the log is 5, which is 2 plus 3, the log of 9 plus the log of 27. That is the product rule:
log_b(MN) = log_b(M) + log_b(N)
It holds for any positive M and N, for the same reason. Logs are exponents: if M is b to the power p and N is b to the power q, then MN is b to the power p + q. So the log of the product is the sum of the logs.
The rule is most useful when one factor is a letter. For positive x, log_3(9x) becomes log_3(9) + log_3(x), and log_3(9) is 2, so it is 2 + log_3(x).
Challenge
Rewrite log_2(8x) with the product rule. Type each line in the box under the problem: first the sum of two logs, then with the number's log worked out. You are done when it reads log_2(x) plus a whole number.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
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