Question

The least value of the expression $2log_{10}x-log_{x} (0.01)$
 is ____

Answer 1

$2log_{10} x-log_{x} (10^{-2})=2(log_{10} x+ \frac{1}{log_{10} x} )$
as we know always
$A.M \geq G.M$
$\frac{log_{10} x+ \frac{1}{log_{10} x}}{2} \geq (log_{10} x \frac{1}{log_{10} x})^{0.5}$
${log_{10} x+ \frac{1}{log_{10} x}} \geq2$
so the minimum value is 4