Question

If $ {\log _{10}}3 = 0.477 $ then, find how many digits are in $ {3^{40}} $ .

Answer 1

Hint: This question is related to logarithm and related concepts. Before solving the given question, we should know what are logarithm functions. Logarithm functions are just the inverse of exponential functions. All the logarithm functions can be converted in exponential form. In order to solve this equation, we will be required to use some of the logarithm function properties.
Formula used:
 $ {\log _b}{a^n} = n{\log _b}a $

Complete step-by-step answer:
Given is $ {\log _{10}}3 = 0.477 $
We are supposed to find the number of digits in $ {3^{40}} $
Now, according to the question
Let,
 $ x = {3^{40}} $
We know the following logarithm power rule:
 $ {\log _b}{a^n} = n{\log _b}a $
Using the above written rule, we will take logarithm on both the sides and we get,
 $ \Rightarrow {\log _{10}}x = {\log _{10}}\left( {{3^{40}}} \right) $
Solving the above equation, we have
 $
   \Rightarrow {\log _{10}}x = 40{\log _{10}}3 \\
   \Rightarrow {\log _{10}}x = 40 \times 0.477 \\
   \Rightarrow {\log _{10}}x = 19.08 \;
  $
Now,
 $
   \Rightarrow x = {10^{19.08}} \\
   \Rightarrow x = {10^{19 + 0.08}} \\
   \Rightarrow x = {10^{19}} \times {10^{0.08}} \;
  $
We know that the value of $ {10^{19}} $ is $ 20 $ and the value of $ {10^{0.08}} $ is less than $ 10 $ .
Therefore, the value of $ x = {10^{19.08}} $ will be $ 20 $ .
Hence,
If $ {\log _{10}}3 = 0.477 $ , then the number of digits in $ {3^{40}} = 20 $ .

Note: Here in this question, we had to use the given value and find the number of digits in an exponent. Students should keep in mind the properties of logarithmic functions. When the base of common logarithm is $ 10 $ then, the base of a natural logarithm is number $ e $ . Students should not forget the rules of exponents while solving exponent and logarithm related questions. Here in the above question, we used the multiplication rule of exponents while solving the exponents. We considered two digits after the decimal point for easy calculation, students can take up to three digits.