Question

hw 9.2: log properties ii
score: 15.17/28 answered: 6/10
question 6
expand each of the following logarithms using the product rule, quotient rule, and/or power rule. do not use the change of base formula.
ln (13w⁸a²) =
log₄ (ay²)⁷ =
question help: video

Explanation:

Step1: Expand $\ln(13w^8a^2)$

Using the Product Rule of Logarithms: $\ln(xyz) = \ln x + \ln y + \ln z$.
So, $\ln(13w^8a^2) = \ln 13 + \ln w^8 + \ln a^2$.
Then, using the Power Rule: $\ln x^n = n\ln x$.
We get $\ln 13 + 8\ln w + 2\ln a$.

Step2: Expand $\log_4(ay^2)^7$

First, use the Power Rule: $\log_b x^n = n\log_b x$.
So, $\log_4(ay^2)^7 = 7\log_4(ay^2)$.
Then, use the Product Rule: $\log_b(xy) = \log_b x + \log_b y$.
We get $7(\log_4 a + \log_4 y^2)$.
Again, using the Power Rule: $7(\log_4 a + 2\log_4 y)$.

Answer:

For $\ln(13w^8a^2)$: $\ln 13 + 8\ln w + 2\ln a$
For $\log_4(ay^2)^7$: $7\log_4 a + 14\log_4 y$ (or $7(\log_4 a + 2\log_4 y)$)