Question

The value of ${{\log }^{0.01}}$ is equal to;

Answer 1

Hint:
For approaching the logarithmic expression the key points are either you solve by approaching the base or by power. Logarithms are the inverse function of exponents so always try to approach by manipulation of power and then try to solve according to some rules. If the base is not defined then we can take it as 10 as we have done in this question.

Complete step by step solution:
According to formula \[\log _y^{{x^n}} = n\log _y^x\]
$\log _{10}^{0.01}$ can be written as :
$\log _{10}^{\dfrac{1}{100}}=\log _{10}^{{{100}^{-1}}}$
     $=\log _{10}^{{{10}^{-2}}}$
So, $\log _{10}^{{{10}^{-2}}}=-2\log _{10}^{10}$
$ = - 2 \times 1 = - 2$ [since, \[\log _y^x = 1\]]

Note:
Always try to change the $\log $ in its base from
$\log _{{{y}^{n}}}^{x}={}^{1}/{}_{n}\log _{y}^{x}=\log _{y}^{{{x}^{{}^{1}/{}_{n}}}}$
Apart from logarithm there exists an inverse process of logarithm which is called anti-logarithms. If x is the logarithm of a number y with a given base b, then y is the antilogarithm of (anti-log) of x to the base b. This method is quite useful in another logarithmic problem. There are some other formulas of log which are:
Product Rule: $\log \,(xy)\,=\,\log (x)+\log (y)$
Quotient Rule: $\log (x/y)\,=\,\log (x)-\log (y)$