Question

expand the logarithm fully using the properties of logs. express the final answer in terms of \\(\log x, \log y,\\) and \\(\log z\\). \\(\log \dfrac{y^2 z^5}{x^4}\\)

Explanation:

Step1: Apply Quotient Rule

The quotient rule of logarithms states that $\log\frac{a}{b}=\log a - \log b$. So, for $\log\frac{y^{2}z^{5}}{x^{4}}$, we get $\log(y^{2}z^{5})-\log(x^{4})$.

Step2: Apply Product Rule

The product rule of logarithms states that $\log(ab)=\log a+\log b$. Applying this to $\log(y^{2}z^{5})$, we have $\log y^{2}+\log z^{5}$. So now the expression is $\log y^{2}+\log z^{5}-\log x^{4}$.

Step3: Apply Power Rule

The power rule of logarithms states that $\log a^{n}=n\log a$. Applying this to each term: $\log y^{2}=2\log y$, $\log z^{5}=5\log z$, and $\log x^{4}=4\log x$. Substituting these back, we get $2\log y + 5\log z-4\log x$.

Answer:

$2\log y + 5\log z - 4\log x$