Question

If \[{\log _{10}}2 = 0.3010....\] the number of digits in the number \[{2000^{2000}}\]is
A.6601
B.6602
C.6603
D.6604

Answer 1

Hint: In this question, since the number whose number of digit is to be found is a large number so we will try to break that number in the logarithmic form and by using the power rule and the product rule of the logarithm we will reduce the number and then by substituting the given values we will find the number of digits for the given number.

Complete step-by-step answer:
Given that \[{\log _{10}}2 = 0.3010....\]
Taking log in the given number whose digits is to be found \[\log \left( {{{2000}^{2000}}} \right)\]
Now we know the logarithm power rule (\[{\log _b}{x^y} = y{\log _b}x\]), hence by using this property we can further write the number as
\[\log \left( {{{2000}^{2000}}} \right) = 2000\log 2000 - - (i)\]
Now factorize 2000, so we can write the number as \[2000 = 2 \times {10^3}\], hence we can further write equation (i) as
\[\Rightarrow \log \left( {{{2000}^{2000}}} \right) = 2000\log \left( {2 \times {{10}^3}} \right)\]
Now since the product rule of logarithm says \[{\log _b}\left( {x \times y} \right) = {\log _b}\left( x \right) + {\log _b}\left( y \right)\], hence we can further write
\[
\Rightarrow \log \left( {{{2000}^{2000}}} \right) = 2000\left[ {\log \left( 2 \right) + \log \left( {{{10}^3}} \right)} \right] \\
   = 2000\left[ {\log \left( 2 \right) + 3\log \left( {10} \right)} \right] \\
 \]
We know \[{\log _{10}}10 = 1\], hence we can write
  \[
\Rightarrow \log \left( {{{2000}^{2000}}} \right) = 2000\left[ {\log \left( 2 \right) + 3\log \left( {10} \right)} \right] \\
   = 2000\left[ {0.3010 + 3} \right] \\
 \]\[\left\{ {\because {{\log }_{10}}2 = 0.3010....} \right\}\]
By further solving
\[
\Rightarrow \log \left( {{{2000}^{2000}}} \right) = 2000\left[ {0.3010 + 3} \right] \\
   = 602 + 6000 \\
   = 6602 \;
 \]
Hence we get the number of digits in the number \[{2000^{2000}}\]is \[ = 6602\]
Now we know when we take log with base 10 then the integer in answer shows one less digit then the actual number of digits in the given number, hence we can say the number of digits in
\[
\Rightarrow {2000^{2000}} = 6602 + 1 \\
   = 6603 \;
 \]
Therefore the number of digits in the number \[{2000^{2000}}\]is \[ = 6603\]
So, the correct answer is “Option C”.

Note: Students must note that when we calculate log for any given number with the base 10, then the integer just greater than that calculated gives the number of digits in the given number.