Evaluate each log without a calculator[tex]log_{4} \frac{1}{16}[/tex][tex]log_{2}\sqrt[5]{32}[/tex]

QUESTION 1The given logarithmic expression is[tex]\log_4(\frac{1}{16})[/tex]We rewrite [tex]\frac{1}{16}[/tex] in the index form to base 4.This implies that;[tex]\log_4(\frac{1}{16})=\log_4(4^{-2})[/tex]We now apply the power rule: [tex]\log_a(m^n)=n\log_a(m^n)[/tex].[tex]\log_4(\frac{1}{16})=-2\log_4(4)[/tex]Recall that logarithm of the base is 1.[tex]\log_4(\frac{1}{16})=-2(1)[/tex][tex]\log_4(\frac{1}{16})=-2[/tex]QUESTION 2The given logarithm is;[tex]\log_2(\sqrt[5]{32})[/tex][tex]\log_2(\sqrt[5]{2^5})[/tex]This is the same as;[tex]\log_2(2^{5\times \frac{1}{5}})[/tex][tex]\log_2(2^{1})[/tex][tex]\log_2(2)=1[/tex]