The Power Rule
Part of the Exponentials, Logs & Sequences section of Coddy's Algebra 2 journey. Lesson 23 of 49.
Take log_2(8^2). Since 8 = 2^3, the power is (2^3)^2, which is 2^6: a power of a power multiplies the exponents. So the log is 6, which is 2 times 3, the exponent times the log of 8. That is the power rule:
log_b(M^k) = k log_b(M)
It holds for positive M, and the exponent comes down in front of the log. The reason: if M is b to the power p, then M to the power k is b to the power kp. You can also see it as the product rule repeated: log_b(M^3) is the log of M times M times M, which is three logs of M added.
With a letter, watch the sign. For x > 0, log_3(x^4) becomes 4log_3(x). At x = -1 the left side is log_3(1), which is 0, but log_3(-1) has no value, so the rule needs x positive. The rule works in every base, base 10 included.
Challenge
For positive x, rewrite log(x^5) with the power rule. Type the result in the box under the problem, then press Check.
Try it yourself
Simplify the expression
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Exponentials, Logs & Sequences
1Exponential Functions
Repeated MultiplyingGrowth or DecayDrawing GrowthDrawing DecayRecap: Exponential Functions