Question

Use the Change of Base Formula to evaluate the expression. Then convert it to a logarithm in base 8. Round to the nearest thousandth if necessary

log5 62

0.390; log8 2.25

2.564; log8 206.93

1.985; log8 24.404

0.025; log8 1.053

Answer 1

`\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} \\\\\\ \begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{ \textit{we'll use this one} }{a^(log_a (x))=x} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}`

`\log_5(62)\implies \cfrac{\log_8(62)}{\log_8(5)} ~~ \approx ~~ 2.56433 ~~ \approx ~~ \boxed{2.564} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{now we can say that}}{2.56433=\log_8(y)}\implies 8^(2.56433)=8^(\log_8(y))\implies 8^(2.56433)=y\implies 206.930\approx y \\\\\\ \textit{so we can write that in logarithmic form as}\qquad \boxed{\log_8(206.93)}`