Question

Given ${\log }_{10}25=x,{\log }_{10}75=y$, evaluate in terms of $x$ and $y$
(i)${\log }_{10}3$
(ii)${\log }_{10}2$

Answer 1

${log}_{10}25=x;{log}_{10}75=y$

(i)$\Rightarrow {log}_{10}3\times 25=y$

$\Rightarrow {log}_{10}3+{log}_{10}25=y$

$\Rightarrow {log}_{10}3=y-{log}_{10}25$

$=y-x$

(ii) $\Rightarrow {log}_{10}25={log}_{10}\frac{100}{4}$

$\Rightarrow x={log}_{10}100-{log}_{10}4$

$\Rightarrow x=2-{2log}_{10}2$

$\Rightarrow {log}_{10}2=\frac{2-x}{2}$

$\left(OR\right)$

${log}_{10}75={log}_{10}\frac{3\times 100}{4}$

$\Rightarrow y={log}_{10}100+{log}_{10}3-{log}_{10}4$

$\Rightarrow y=2+y-x-2{log}_{10}2$

$\Rightarrow {log}_{10}2=\frac{2-x}{2}$