Question

If 10^0.3010 = 2, then find the value of log0.125 (125) ?

Answer 1

Given,
10^0.3010 = 2

So,
$log_{10}(2) = 0.3010$
Now,

$log_{0.125}(125) = \frac{ log(125) }{ log(0.125) } \\ = \frac{ log( {5}^{3} ) }{ log( {0.5}^{3} ) } \\ = \frac{3}{3 } \frac{ log(5) }{ log( \frac{1}{2} ) } \\ = \frac{ log(5) }{ log(1) - log(2) } \\ = - \frac{ log(5) }{ log(2) } \\ = - 2.321$

Hope helped!

Answer 2

$\bold{10^{0.3010}=2}$
               take log base 10 both sides,
$\bold{log_{10}{10^{0.3010}}=log_{10}2}\\\\\bold{0.3010log_{10}10=log_{10}2}\\\\\bold{0.3010=log_{10}2}$

now, given,
 $\bold{log_{0.125}125}=\bold{log_{\frac{125}{1000}}125}\\\\=\bold{log_{0.5}5}=\bold{\frac{1}{log_50.5}}=\bold{\frac{1}{-log_52}}\\\\=\bold{\frac{-log_{10}5}{log_{10}2}}$
 = -2.321  [ log5 = 0.6989 and log2 = 0.3010]