Question

Given $lo{g}_{10}25=x,lo{g}_{10}75=y$, evaluate without using logarithmic tables, in terms of $x,y$
(i) $log{10}^3,$
(ii) $log{10}^2$

Answer 1

Given that

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

then

$\log 25=x,\log 3\times 25=y$

$\log 5^2=x;\log 3+\log 25=y$

$2\log 5=x;\log 3+x=y$

$\log 5=x/2;\log 3=y-x[\because \log a\times b=\log a+\log b]$

$\log 2=\log \left(\frac{10}{5}\right)={\log }_{10}10-\log 5$

$\log 2=1-\frac{x}{2}$

(i) $\log {10}^3=\log 1000=\log 250\times 4=\log 25+{\log }_{10}10+\log 4=\log 5^2+1+2\log 2=x+1+2(1-\frac{x}{2})$

$\log {10}^3=x+1+2-x=3$

(ii) $\log {10}^2=2\log 10=2\times {\log }_{10}10=2\times 1=2[\because \log 10=1]$