Question

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

Answer 1

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.