Explanation
We are given the following:
[tex]\begin{gathered} \log_bA=3 \\ \operatorname{\log}_bC=2 \\ \operatorname{\log}_bD=5 \end{gathered}[/tex]
We are required to determine the value of:
[tex]\operatorname{\log}_b(\frac{A^5C^2}{D^6})[/tex]
3We know that the multiplication and division laws of logarithm state:
Therefore, we have:
[tex]\begin{gathered} \log_b(\frac{A^5C^2}{D^6})=\log_b(A^5C^2)-\log_bD^6 \\ =\log_bA^5+\log_bC^2-\log_bD^6 \end{gathered}[/tex]
We also know that the power law of logarithm states:
Therefore, we have:
[tex]\begin{gathered} \Rightarrow\log_bA^5+\log_bC^2-\log_bD^6 \\ =5\log_bA+2\log_bC-6\log_bD \\ Substituting\text{ }their\text{ }values \\ =5(3)+2(2)-6(5) \\ =15+4-30 \\ =-11 \end{gathered}[/tex]
Hence, the answer is -11.
