Question

Find the value of x if \[{{\log }_{100}}x=-4\]
(A) \[\dfrac{1}{{{10}^{-4}}}\]
(B) \[{{10}^{-4}}\]
(C) \[\dfrac{1}{{{10}^{8}}}\]

Answer 1

Hint: We know that if \[{{\log }_{b}}a=c\] then, the value of a is equal to \[{{b}^{c}}\] . Now, compare the equations \[{{\log }_{b}}a=c\] and \[{{\log }_{100}}x=-4\] . Then, get the values of a, b, and c. Replace a by x, b by 100, and c by -4, in the equation \[a={{b}^{c}}\] . We know that 100 can be written as the square of 10. We also know that \[{{a}^{-n}}\] can be written as \[\dfrac{1}{{{a}^{n}}}\] . Now, solve it further and get the value of x.

Complete step by step solution:
According to the question, it is given that the value of \[{{\log }_{100}}x\] is equal to -4. We can also write it as,
 \[{{\log }_{100}}x=-4\] …………………….(1)
We know that if \[{{\log }_{b}}a=c\] then, the value of a is equal to \[{{b}^{c}}\] . We can write it as,
\[a={{b}^{c}}\] ……………………………….(2)
Now, comparing the equation \[{{\log }_{100}}x=-4\] and \[{{\log }_{b}}a=c\] , we get
The value of a is equal to x.
\[a=x\] ……………………….(3)
The value of b is equal to 100.
\[b=100\] …………………………(4)
The value of c is equal to -4.
\[c=-4\] …………………………(5)
Now, using equation (2) to get the value of x.
From equation (3), equation (4), and equation (5), we have the values of a, b, and c.
Replacing a by x, b by 100, and c by -4, in equation (2), we get
\[x={{100}^{-4}}\] ………………………..(6)
We know that 100 can be written as the square of 10, \[100={{10}^{2}}\] ………………………(7)
Now, from equation (6) and equation (7), we get
\[x={{\left( {{10}^{2}} \right)}^{-4}}\] ………………………(8)
We know the formula, \[{{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}\] ………………….(9)
Now, using the formula shown in equation (9) and simplifying equation (8), we get
\[\Rightarrow x={{\left( 10 \right)}^{2\times \left( -4 \right)}}\]
\[\Rightarrow x={{\left( 10 \right)}^{-8}}\] ………………………….(10)
We also know that \[{{a}^{-n}}\] can be written as \[\dfrac{1}{{{a}^{n}}}\] , \[{{a}^{-n}}=\dfrac{1}{{{a}^{n}}}\] …………………………….(11)
From equation (10) and equation (11), we get
\[\Rightarrow x=\dfrac{1}{{{10}^{8}}}\] .
So, the value of x is \[\dfrac{1}{{{10}^{8}}}\] .
Therefore, the correct option is (C).

Note: In this question, after replacing a by x, b by 100, and c by -4, in the equation \[a={{b}^{c}}\] , we get \[x={{100}^{-4}}\] . Here, one might make a silly mistake and go with option (B). But this is wrong. Since option (B) has \[{{10}^{-4}}\] and we have got the value of x equal to \[{{100}^{-4}}\] so, we cannot go with option (B).