Question

use the properties of logarithms to expand \\(\log \frac{6^z}{x}\\). each logarithm should involve only one variable and should not have any exponents or fractions. assume that all variables are positive.

Explanation:

Step1: Apply Quotient Rule

The quotient rule of logarithms states that $\log_b \frac{M}{N} = \log_b M - \log_b N$. For $\log \frac{6^z}{x}$, we have $M = 6^z$ and $N = x$. So, $\log \frac{6^z}{x}=\log 6^z - \log x$.

Step2: Apply Power Rule

The power rule of logarithms states that $\log_b M^n = n\log_b M$. For $\log 6^z$, using the power rule, we get $z\log 6$. So the expanded form is $z\log 6 - \log x$.

Answer:

$z\log 6 - \log x$