Solve the expression \[\dfrac{{\left( {66.66 \times 36.36} \right)}}{{26.26}}\] using Logarithm table (Clark’s tables).

Hint: Logarithm converts complex problems of multiplication and division into easier addition and subtraction problems.
Logarithm is an inverse function of an exponentiation. It is a mathematical operation that determines how many times the base is multiplied by itself to reach another number. It is the power to which a number is raised to get another number.
An antilog is the inverse process of finding the logarithm of a same number with base b.
Some of the properties of the Logarithm:-
 \[{\log _a}mn = {\log _a}m + {\log _a}n\] ; It is known as product rule law. It means the logarithm of the product of two or more positive factors to any positive base other than \[1\] is equal to the sum of the logarithm of the factor to the same base.
 \[{\log _a}\dfrac{m}{n} = {\log _a}m - {\log _a}n\] ; It is known as quotient rule law. It means the logarithm of the quotient of two factors to any positive base other than \[1\] is equal to the difference of the logarithm of the factor to the same base.
Logarithm table is used to find the value of a logarithm function .These tables contain common logarithms of base 10 \[({\log _{10}})\] which were extensively being used in computation.

Complete step-by-step solution:
We have to solve the given problem,
 \[\dfrac{{\left( {66.66 \times 36.36} \right)}}{{26.26}}\]
Let \[a = \dfrac{{\left( {66.66 \times 36.36} \right)}}{{26.26}}\]
Taking \[\log {}_{10}\] both sides, we get,
\[\log {}_{10}a = {\log _{10}}\dfrac{{\left( {66.66 \times 36.36} \right)}}{{26.26}}\]
After applying quotient rule law, we get
\[ \Rightarrow \log {}_{10}a = {\log _{10}}\left( {66.66 \times 36.36} \right) - {\log _{10}}26.26\] ; since \[{\log _a}\dfrac{m}{n} = {\log _a}m - {\log _a}n\]
Here \[m\] represent \[\left( {66.66 \times 36.36} \right)\] and \[n\] represent \[26.26\] .
After applying product rule law, we get,
 \[ \Rightarrow {\log _{10}}a = {\log _{10}}66.6 + {\log _{10}}36.36 - {\log _{10}}26.26\] ; since \[{\log _a}mn = {\log _a}m + {\log _a}n\]
Here \[m\] represent \[66.66\] and \[n\] represent \[36.36\] .
 \[ \Rightarrow {\log _{10}}a = 1.82 + 1.5606 - 1.4193\] ;
Since \[{\log _{10}}66.66 = 1.82\] ; \[{\log _{10}}36.36 = 1.5606\] ; \[{\log _{10}}26.26 = 1.4193\]
 \[ \Rightarrow {\log _{10}}a = 1.9613\]
Applying \[Anti\log \] both sides,
 \[ \Rightarrow a = Anti\log \left( {1.9613} \right)\]
 \[ \Rightarrow a = 91.47\] ;
Since \[Anti\log \left( {1.9613} \right) = 91.47\]
Thus we get,
 \[\dfrac{{\left( {66.66 \times 36.36} \right)}}{{26.26}} = 91.47\]

Note: In this sum , we have to use Product Rule Law and Quotient rule law to get the answer . We must keep in mind that Product rule law is true for more than two positive factors. Here we generally use the logarithm of base 10 \[({\log _{10}})\]. But we need to know about the natural log represented by \[\ln \] , it is the base ‘e’ log \[({\log _e})\] where \[e{\text{ }} = {\text{ }}2.718\] , is a constant. ‘e’ is sometimes known as natural number or Euler’s number.We should try to solve the problem using \[({\log _{10}})\] , as it is easy to use and Log table uses base 10 logs \[({\log _{10}})\] which is also called the common logarithms.