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# How do you evaluate $\log \,0.01$? Verified
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Hint: Logarithm of the form $\log \,a$ has a base of the logarithm as 10. In logarithm we have several properties like $\log \,\dfrac{m}{n}\,\,=\,\,\log m\,-\,\log n$ (where $m$,$n$ are positive numbers)
${{\log }_{a}}a\,\,=\,\,1$, $\log 1\,=\,0\,$ etc.

Complete step by step solution:
Definition of logarithm:
Every positive real number $N$ can be expressed in exponential form as ${{a}^{x}}\,\,=\,\,N$where ' $a$' is also a positive real number different than unity and is called the base and ' $x$' is called an exponent. We can write the relation ${{a}^{x}}\,\,=\,\,N$in logarithmic form as ${{\log }_{a}}N\,=\,x$. Hence ${{a}^{x}}\,\,=\,\,N\,\,\Leftrightarrow \,\,{{\log }_{a}}N\,=\,x$.
Hence, the logarithm of a number to some base is the exponent by which the base must be raised in order to get that number.
Limitations of logarithm: ${{\log }_{a}}N\,=\,x$ is defined only when (i)$N\,>\,0$, (ii) $a\,>\,0$ (iii) $a\,\ne \,1$
We can evaluate $\log \,0.01$ step by step by transforming it in such a form so that we can apply the known formulae or properties of logarithm.
$\log \,0.01$ can be written as $\log \left( \dfrac{1}{100} \right)$.
As $\log \,\dfrac{m}{n}\,\,=\,\,\log m\,-\,\log n$.
$\,\Rightarrow \,\,\log \left( \dfrac{1}{100} \right)\,\,=\,\,\log 1\,-\,\log 100\,=\,0\,-\,\log {{10}^{2}}$ ……………………………………………………… (i)
also as $\log 1\,=\,0$ and $\log {{a}^{n}}\,=\,n\times \log a$(power rule of logarithm) equation (i) reduces to
$\,\Rightarrow \,\,\log \left( \dfrac{1}{100} \right)\,\,=\,\,0\,\,-\,\,2\times \log (10)\,\,=\,\,0\,\,-\,\,2\times 1$
$\therefore \,\,\,\,\log 0.01\,\,=\,\,-2$.

Note:
> $\log a$ has the base of the logarithm as 10 whereas $\log a$ has the base of the logarithm as $e$, where $e$ is Napier’s constant. Napier’s constant is an irrational number. The approximate value of Napier’s constant is $e\,\,=\,\,2.718$.
> For a given value of $N$, ${{\log }_{a}}N$ will give us a unique value.
> Logarithm of zero does not exist.
${{\log }_{N}}N\,=\,\,1$
> Logarithms of negative real numbers are not defined in the system of real numbers.