Question

The number of digits in \[{20^{301}}\] ( given , \[{\log _{10}}2 = 0.3010\]) is
A.\[602\]
B.\[301\]
C.\[392\]
D.\[391\]

Answer 1

Hint: In the question we have given the value of logarithm \[2\] , therefore we must use the concept of logarithm to solve it . We will use the product law for logarithm and exponent law of logarithm to solve it . The number of digits will always be one greater than the characteristic. Always use the given value of logarithm in the question . Remember the laws for logarithm and exponent . In a logarithm number the value left of the decimal place is called characteristic and the value right of the decimal is called mantissa .

Complete step-by-step answer:
Given : \[{20^{301}}\] , \[{\log _{10}}2 = 0.3010\]
Let \[x = {20^{301}}\]
Taking \[\log \] on both sides we get , here we have used base \[10\] .
\[\log x = \log {20^{301}}\]
Now using the laws of exponent for logarithm \[\log {m^n} = n\log m\] on LHS we get ,
\[\log x = 301\log 20\]
On simplifying we get ,
\[\log x = 301\log \left( {2 \times 10} \right)\]
Now using the product law for logarithm \[\log m \times n = \log m + \log n\] we get ,
\[\log x = 301\left[ {\log 2 + \log 10} \right]\] ,
Now using the given value of \[{\log _{10}}2 = 0.3010\] and \[{\log _{10}}10 = 1\] in the above equation , we get
\[\log x = 301\left[ {0.3010 + 1} \right]\]
On solving we get ,
\[\log x = 301\left[ {1.3010} \right]\]
On solving again we get ,
\[\log x = {\text{391}}{\text{.601}}\]
The characteristic part has value \[ = 391\]
According to hint , the required value will be one greater than , therefore \[ = 391 + 1\] ,
\[ = 392\]
Therefore , option ( 2 ) is the correct answer for the given question .
So, the correct answer is “Option 2”.

Note: The question can be calculated using integration method which is as follow :
Number of digits = integrated part of \[\left( {301\log 20} \right) + 1\]
On solving we get ,
integrated part of \[301\log \left( {2 \times 10} \right) + 1\]
integrated part of \[301\left[ {\log 2 + \log 10} \right] + 1\]
integrated part of \[301\left[ {1.3010} \right] + 1\]
integrated part of \[\left( {{\text{391}}{\text{.601}}} \right){\text{ + 1}}\]
\[ = 391 + 1\]
\[ = 392\]
Hence Proved .