Answer:
[tex]\log _52+2\log _5x^{}-3\log y[/tex]
Explanation:
Given the logarithm expression:
[tex]\log _5\mleft(\frac{2 x^{2}}{y^{3}}\mright)[/tex]
To expand the expression, follow the steps below:
Step 1: Apply the division law of logarithm below. That is, the log of a quotient is the difference between the logs. Therefore:
[tex]\begin{gathered} \log (\frac{A}{B})=\log (A)-\log (B) \\ \implies\log _5\mleft(\frac{2 x^{2}}{y^{3}}\mright)=\log _5(2x^2)-\log (y^3) \end{gathered}[/tex]
Step 2: Similarly, by the multiplication law, the log of a product is the sum of the logs.
[tex]\begin{gathered} \log (AB)=\log (A)+\log (B) \\ \log _5(2x^2)=\log _52+\log _5x^2 \\ \implies\log _5(2x^2)-\log (y^3)=\log _52+\log _5x^2-\log (y^3) \end{gathered}[/tex]
Step 3: We apply the index law of logarithm.
If the number whose logarithm we are looking for has an index (or power), we can write the index as a product.
[tex]\log x^n=n\log x[/tex]
So, we have that:
[tex]\implies\log _5(2x^2)-\log (y^3)=\log _52+2\log _5x^{}-3\log y[/tex]
Thus, the expanded form of the given expression is:
[tex]\log _5\mleft(\frac{2 x^{2}}{y^{3}}\mright)=\log _52+2\log _5x^{}-3\log y[/tex]
