Question

expand the logarithm fully using the properties of logs. express the final answer in terms of log x, and log y. \\(\log \frac{x^{3}}{y^{5}}\\)

Explanation:

Step1: Apply Quotient Rule of Logs

The quotient rule of logarithms states that $\log \frac{a}{b} = \log a - \log b$. For $\log \frac{x^3}{y^5}$, we have:
$\log \frac{x^3}{y^5} = \log x^3 - \log y^5$

Step2: Apply Power Rule of Logs

The power rule of logarithms states that $\log a^n = n\log a$. Applying this to both terms:
For $\log x^3$, we get $3\log x$.
For $\log y^5$, we get $5\log y$.

Step3: Combine the Results

Substituting the results from Step 2 into the equation from Step 1, we have:
$\log \frac{x^3}{y^5} = 3\log x - 5\log y$

Answer:

$3\log x - 5\log y$