Question

Please help! it’s about logarithms and logarithmic functions

Answer 1

`\textit{exponential form of a logarithm} \\\\ \log_a(b)=y \qquad \implies \qquad a^y= b \\\\\\ \textit{Logarithm Change of Base Rule} \\\\ \log_a b\implies \cfrac{\log_c b}{\log_c a}\qquad \qquad c= \begin{array}{llll} \textit{common base for }\\ \textit{numerator and}\\ denominator \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}`

`f(x)=\log_{(1)/(4)}(x)\hspace{5em}y=\log_{(1)/(4)}(x) \\\\[-0.35em] ~\dotfill\\\\ f(16)\implies y=\log_{(1)/(4)}(16)\implies \left( \cfrac{1}{4} \right)^y=16\implies \left( \cfrac{1}{4} \right)^y=4^2 \\\\\\ (4^(-1))^y=4^2\implies (4)^(-y)=4^2\implies -y=2\implies \boxed{y=-2} \\\\[-0.35em] ~\dotfill`

`f\left( (1)/(64) \right)\implies y=\log_{(1)/(4)}\left( (1)/(64) \right)\implies \left( \cfrac{1}{4} \right)^y=\cfrac{1}{64}\implies \left( \cfrac{1}{4} \right)^y=\cfrac{1^3}{4^3} \\\\\\ \left( \cfrac{1}{4} \right)^y=\left( \cfrac{1}{4} \right)^3\implies \boxed{y=3} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)=\log_{(1)/(4)}(x)\implies f(x)=\cfrac{\log(x)}{\log\left( (1)/(4) \right)}\qquad \impliedby \textit{plug that in the calculator}`

Check the picture below.