Question

if log10 25= a and log10 75=b express the values of log10 9 and log10 4 in terms of and b​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\: log_{10}(25) = a$

$\rm :\longmapsto\: log_{10}( {5}^{2} ) = a$

$\rm :\longmapsto\: 2log_{10}( {5}) = a$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \bigg \{ \because \: log( {x}^{y} ) = y \: logx \bigg \}}$

$\bf\implies \: log_{10}(5) = \dfrac{a}{2} - - - (1)$

Also,

$\rm :\longmapsto\: log_{10}(75) = b$

$\rm :\longmapsto\: log_{10}(3 \times {5}^{2} ) = b$

$\rm :\longmapsto\: log_{10}(3) + log_{10}( {5}^{2} ) = b$

$\: \: \: \: \: \: \: \: \: \: \: \: \red{ \bigg \{ \because \: log(xy) = logx + logy\bigg \}}$

$\rm :\longmapsto\: log_{10}(3) + 2 log_{10}( {5}) = b$

$\rm :\longmapsto\: log_{10}(3) = b - 2 log_{10}( {5})$

$\bf\implies \: log_{10}(3) = b - a$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \bigg \{ \because \: 2log_{10}(5) = a \bigg \}}$

Consider,

$\red{\bf :\longmapsto\: log_{10}(9)}$

$\rm \: \: = \: \: log_{10}( {3}^{2} )$

$\rm \: \: = \: \: 2 \: log_{10}(3)$

$\rm \: \: = \: \: 2 \: (b - a)$

$\: \: \: \: \: \: \: \: \: \: \: \boxed{\red{\bf :\implies\: log_{10}(9) = 2(b - a)}}$

Consider,

$\green{\bf :\longmapsto\: log_{10}(4)}$

$\rm \: \: = \: \: log_{10}( {2}^{2} )$

$\rm \: \: = \: \: 2 \: log_{10}(2)$

$\rm \: \: = \: \: 2 \: log_{10}\: \bigg(\dfrac{10}{5} \bigg)$

$\rm \: \: = \: \: 2 \: \bigg( log_{10}(10) - log_{10}(5) \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \red{ \bigg \{ \because \: log\dfrac{x}{y} = logx - logy \bigg \}}$

$\rm \: \: = \: \: 2 \bigg (1 - \dfrac{a}{2} \bigg)$

$\rm \: \: = \: \: 2 - a$

$\: \: \: \: \: \: \: \: \: \: \: \boxed{\green{\bf :\implies\: log_{10}(4) = 2 - a}}$