Question

the value of log25 base8 given log2 = 0.3010
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Answer 1

$\bf\large\underline{\underline{\green {Solution}:-}}$

$\rm\underline{Given:-}$

• $\sf\log2 = 0.3010$

$\rm\underline{We \; need\;to\;find:}$

• $\sf\log_8{25}$

$\sf Now , \\\\\sf \longrightarrow log_8 25\\\\\small\rm\diamond\; Substituting\;25=100/4\\\\\sf\longrightarrow log_{8}{100/4}\\\\\rm\small \diamond\; Using\; log_{n}{a/b}=log_n a -log_n b \\\\\sf\longrightarrow log_8 10^2 - log_8 2^2\\\\ \small\rm \diamond\; Using \; loga^m = mloga\\\\\sf\longrightarrow 2log_{2^3} 10 - 2log_{2^3}2\\\\\small\rm\diamond \;Using \;log_{b^n}{a^m} = \dfrac{m}{n}log_b a\\\\\sf\longrightarrow 2/3(log_2 10-log_2 2)\\\\\diamond\small\rm \;Using \;log_a a = 1 \; \& \; log_b a = \dfrac{1}{log_a b}\\\\\longrightarrow \sf 2/3\bigg(\dfrac{1}{log2}-1\bigg)= 2/3\bigg(\dfrac{1-log2}{2}\bigg)\\\\\diamond\small\rm Substituting\; log2=0.3010\\\\\sf\longrightarrow 2/3\bigg(\dfrac{1-0.3010}{0.3010}\bigg)=2/3(0.699/0.3010)\\\\\small\sf\longrightarrow 2/3(2.32)\\\\\huge\bf\dagger\boxed{\purple{\bf 1.5466}}$