
How do you evaluate \[\log \,0.01\]?
Answer
408k+ views
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.
\[{{\log }_{a}}a\,\,=\,\,1\], \[\log 1\,=\,0\,\] 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.
Recently Updated Pages
Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Statement I Reactivity of aluminium decreases when class 11 chemistry CBSE

Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Difference Between Plant Cell and Animal Cell

What is pollution? How many types of pollution? Define it

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.
