Question

What is the value of ${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$?
(a) 4
(b) 3
(c) 2
(d) 1

Answer 1

Hint: To find the value of the given expression we are going to use the property of the logarithm. The expression given above is as follows: ${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$. Now, we can write 100 as ${{10}^{2}}$ in this expression and we get ${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}{{10}^{2}} \right) \right]}^{2}}$. After doing this, we are going to use the property of the logarithm which says that: ${{\log }_{a}}{{a}^{n}}=n$. Then the same property we are going to use again and simplify the given expression.

Complete step-by-step answer:
The logarithm expression given in the above problem is as follows:
${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$
In the above expression we are going to write 100 as ${{10}^{2}}$ then the above expression will look as follows:
${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}{{10}^{2}} \right) \right]}^{2}}$ …………. (1)
We know the logarithm property which is written as follows:
${{\log }_{a}}{{a}^{n}}=n$
In the above equation, “n” is the exponent for $''a''$. Now, we are going to apply the above property in ${{\log }_{10}}{{10}^{2}}$ then $a=10$ and $n=2$ so substituting these values in the above equation we get,
${{\log }_{10}}{{10}^{2}}=2$
Using the above relation in (1) we get,
$\begin{align}
  & {{\left[ {{\log }_{10}}\left( 5\left( 2 \right) \right) \right]}^{2}} \\
 & ={{\left[ {{\log }_{10}}\left( 10 \right) \right]}^{2}}............\left( 2 \right) \\
\end{align}$
Now, again we can use the logarithm property which we have shown in the above solution where $a=10$ and $n=1$ so substituting these values in ${{\log }_{a}}{{a}^{n}}=n$ we get,
${{\log }_{10}}{{10}^{1}}=1$
Using above relation in (2) we get,
$\begin{align}
  & {{\left[ {{\log }_{10}}\left( 10 \right) \right]}^{2}} \\
 & ={{\left[ 1 \right]}^{2}} \\
 & =1 \\
\end{align}$
From the above solution, we got the value of the given expression as 1.

So, the correct answer is “Option d)”.

Note: To solve the above problem, we need to know the property of logarithm if we don’t know this property then it would become very hard to solve this problem. You may think how we have come to know that we should use this property of the logarithm despite there are so many properties in the logarithm. The answer to this thing is that if you look at the expression carefully then you will find that:
${{\left[ {{\log }_{10}}\left( 5{{\log }_{10}}100 \right) \right]}^{2}}$
In the above expression if we write 100 as ${{10}^{2}}$ then we can use the logarithm property ${{\log }_{a}}{{a}^{n}}=n$.