Question

expand the logarithm fully using the properties of logs. express the final answer in terms of \\(\log x\\), and \\(\log y\\).\\(\log \dfrac{x^3}{y^4}\\)

Explanation:

Step1: Apply the quotient rule of logarithms

The quotient rule states that $\log \frac{a}{b} = \log a - \log b$. So for $\log \frac{x^3}{y^4}$, we have:
$\log x^3 - \log y^4$

Step2: Apply the power rule of logarithms

The power rule states that $\log a^n = n\log a$. Applying this to both terms:
For $\log x^3$, we get $3\log x$.
For $\log y^4$, we get $4\log y$.

Step3: Combine the results

Substituting back, we have $3\log x - 4\log y$.

Answer:

$3\log x - 4\log y$