Question

If \[\log 1001 = 3.00434\] Then find the number of digits in the term, \[{1001^{101}}\]?

Answer 1

Hint: Here in the given question we need to solve by solving the “log” value given in the question, on simplifying with the expression we can reach to the solution we need. By using the “log” property, here after solving we can count the digits and then solve for the given question.
Formulae Used:
\[
   \Rightarrow \log {a^b} = b(\log a) \\
   \Rightarrow if\,\log a = b \to a = {10^b} \\
 \]

Complete step by step answer:
 In order to solve the given question, here we are provided with a “log” value, and with the help of which we can get the solution for the answer. To solve the question here we need to take the “log” of the given value, and then using the properties of “log” function we can get the solution for the question, on solving we get:
On solving with the help of the value given, and then adjusting we get:
\[ \Rightarrow \log 1001 = 3.00434\]
Multiplying by \[101\] on both side we get:
\[
   \Rightarrow 101 \times \log 1001 = 3.00434 \times 101 \\
   \Rightarrow \log \left( {{{1001}^{101}}} \right) = 3.00434 \times 101 \\
   \Rightarrow {1001^{101}} = {10^{3.00434 \times 101}} \\
   \Rightarrow {1001^{101}} = {10^{303.43834}} \\
 \]
Now counting the number of digits in the solution we can see that numbers of digits should be greater than 303.

Note: In order to solve the given question here we need to see the equation given in the question, here we are provided with the “log” equation and then accordingly we do adjustment with the equation so as to obtain the value of the question.