Question

Find the values of x in each of the following:

$\left(i\right)\frac{log144}{log12}=logx$

$\left(ii\right)\frac{log125}{log25}=x$

$\left(iii\right)lo{g}_{x}4+lo{g}_{x}16+lo{g}_{x}64=12$

Answer 1

(i) We have,
$\frac{log144}{log12}=logx$

$\Rightarrow \frac{log{12}^2}{log12}=logx$

$\Rightarrow \frac{2log12}{log12}=logx$

$\Rightarrow logx=2$

$\Rightarrow lo{g}_{10}x=2[\because logx=lo{g}_{10}x]$

$\Rightarrow x={10}^2=100$

(ii) We have

$\frac{log125}{log25}=x$

$\Rightarrow \frac{log{5}^3}{log{5}^2}=x$

$\Rightarrow \frac{3log5}{2log5}=x$

$\Rightarrow \frac{3}{2}=x$

(iii) We have,

$lo{g}_{x}4+lo{g}_{x}16+lo{g}_{x}64=12$

$\Rightarrow lo{g}_x{2}^2+lo{g}_x{2}^4+lo{g}_x{2}^6=12$

$\Rightarrow 2lo{g}_{x}2+4lo{g}_{x}2+6lo{g}_{x}2=12$

$\Rightarrow 12lo{g}_{x}2=12$

$\Rightarrow lo{g}_{x}2=1$

$\Rightarrow x^1=2$

$\Rightarrow x=2$