Question

Find the number of zeros after the decimal in $$3^{-100}$$. If $$\log_{10} 3=0.4771$$

Answer 1

Hint: In this question it is given that if $$\log_{10} 3=0.4771$$, then find the number of zeros after the decimal in $$3^{-100}$$. So to find this first of all we have to consider $t=3^{-100}$, and after that by applying $\log$, we will get the solution.
So in order to get the solution we need some basic formulas of logarithm, which are,
$$\log a^{b}=b\log a$$.........(1)
if $$\log_{a} b=k$$, then $$b=k^{a}$$........(2)
Complete step-by-step solution:
Let, $t=3^{-100}$
Now taking logarithm in the both side of the above equation, we get,
$$\log_{10} t=\log_{10} 3^{-100}$$
$$\Rightarrow \log_{10} t=-100\log_{10} 3$$ [ since, $$\log a^{b}=b\log a$$]
$$\Rightarrow \log_{10} t=-100\times 0.4771$$ [ given, $$\log_{10} 3=0.4771$$]
$$\Rightarrow \log_{10} t=-47.71$$
As we know that, if $$\log_{a} b=k$$ then $$b=a^{k}$$.
So by using the above formula, we get,
$$\log_{10} t=-47.71$$
$$\Rightarrow t=10^{-47.71}$$
Therefore, we can say that the number of zeroes in $$3^{-100}$$ after decimal is 47.
Note: While solving, you need to know the basic formulas of logarithm that we have already mentioned in the hint portion also you have to know that $10^{-n}$ is defines a decimal number where the number of zeros after decimal point is ‘n’ and if ‘n’ is itself a decimal number then the number of zeros will be the greatest integral value which is less than ‘n’.