Question

Find out the number of cyphers after decimal before a significant figure comes in ${{\left( \dfrac{5}{3} \right)}^{-100}}$. [given: ${{\log }_{10}}5=0.6990$, ${{\log }_{10}}3=0.4770$]
A. $21$ B. $22$ C. $23$ D. $24$

Answer 1

Hint: Here cypher means the number of zeroes that come before a significant number comes in a decimal figure. Significant number can be any non-zero digit.
We need to solve the given problem using the formulas of logarithm.
We take the logarithmic value of the given figure.
We will take the help of the given data of particular log values of $3$ and $5$ to get the solution.
Then matching with the given options, we find the final answer.

Complete step-by-step answer:
We have been given the number ${{\left( \dfrac{5}{3} \right)}^{-100}}$ whose number of cyphers we need to find out.
The given number is a very small number whose exact value is hard to find out.
So, we take the help of logarithm.
Let’s assume $p={{\left( \dfrac{5}{3} \right)}^{-100}}$.
Taking logarithm (base 10) both sides we get, ${{\log }_{10}}p={{\log }_{10}}{{\left( \dfrac{5}{3} \right)}^{-100}}$.
Now we know the formulas of logarithm like $\log \left( {{a}^{m}} \right)=m\times \log a$, $\log \left( \dfrac{a}{b} \right)=\log a-\log b$.
So, the given equation becomes ${{\log }_{10}}p={{\log }_{10}}{{\left( \dfrac{5}{3} \right)}^{-100}}=\left( -100 \right){{\log }_{10}}\left( \dfrac{5}{3} \right)$.
Using the second formula we get, ${{\log }_{10}}p=\left( -100 \right)\left[ {{\log }_{10}}5-{{\log }_{10}}3 \right]$.
Now, we use the given particular log values of $3$ and $5$ to get the solution.
Given that, ${{\log }_{10}}5=0.6990$, ${{\log }_{10}}3=0.4770$.
The solution becomes ${{\log }_{10}}p=\left( -100 \right)\left[ 0.699-0.477 \right]=\left( -100 \right)\left[ 0.222 \right]=-22.2$.
We know that ${{\log }_{a}}b=x\Rightarrow {{a}^{x}}=b$.
So, the equation changes into its decimal form by $p={{10}^{-22.2}}=\dfrac{1}{{{10}^{22.2}}}$.
So, the digit number in front of the logarithmic value gives us the cypher number as p has that many zeros after decimal before a significant number comes.
So, for $-22.2$ the cypher number is $22$.
The number of cyphers after decimal before a significant figure comes in ${{\left( \dfrac{5}{3} \right)}^{-100}}$ is $22$.
So, (B) is the correct option.

Note: We need to remember that the cypher number can be found from the logarithmic calculation but to find the actual value or to visualise it we need to remove the logarithm.
Although for getting the answer we don’t need to solve the logarithm. The number after decimal in the logarithmic value is insignificant in getting the answer. We don’t need to show the calculation to get the actual answer.