Question

1. condense each logarithmic expression: a. \\(\frac{4}{3}\log_{b} 8 - 2\log_{b} 4\\)

Explanation:

Step1: Apply the power rule of logarithms ($n\log_b M = \log_b M^n$)

For the first term: $\frac{4}{3}\log_b 8 = \log_b 8^{\frac{4}{3}}$
Calculate $8^{\frac{4}{3}}$: $8 = 2^3$, so $(2^3)^{\frac{4}{3}} = 2^{3\times\frac{4}{3}} = 2^4 = 16$
For the second term: $2\log_b 4 = \log_b 4^2 = \log_b 16$

Step2: Substitute the results back into the original expression

The original expression $\frac{4}{3}\log_b 8 - 2\log_b 4$ becomes $\log_b 16 - \log_b 16$

Step3: Apply the quotient rule of logarithms ($\log_b M - \log_b N = \log_b \frac{M}{N}$)

$\log_b 16 - \log_b 16 = \log_b \frac{16}{16} = \log_b 1$

Step4: Recall that $\log_b 1 = 0$ for any base $b > 0, b

eq 1$
So the condensed form is $0$

Answer:

$0$