Question

If \[{\log _a}\left( {\sqrt x } \right) = 4,\] then the value of ‘a’ is
A. \[{x^4}\]
B. \[{x^{\dfrac{1}{4}}}\]
C. \[{x^{\dfrac{1}{2}}}\]
D. \[{x^{\dfrac{1}{8}}}\]

Answer 1

Hint: This question is based on the log properties the main property we will be using here is, if \[{\log _a}x = n\] then \[x = {a^n}\] so from here we can easily find the value of a.

Complete step by step answer:
So we are given that \[{\log _a}\left( {\sqrt x } \right) = 4,\]
Which can also be written as \[{\log _a}{x^{\dfrac{1}{2}}} = \dfrac{1}{2}{\log _a}x = 4\]
I have basically used the property of log where \[{\log _b}\left( {{M^p}} \right) = p{\log _b}M\]
\[\begin{array}{l}
\therefore \dfrac{1}{2}{\log _a}x = 4\\
 \Rightarrow {\log _a}x = 8\\
 \Rightarrow x = {a^8}
\end{array}\]
So by taking \[{8^{th}}\] root in both the sides we are getting
\[a = {x^{\dfrac{1}{8}}}\]

So, the correct answer is “Option D”.

Note: There are many other log properties we need to remember other than \[{\log _b}\left( {{M^p}} \right) = p{\log _b}M\] some of the important ones are
\[\begin{array}{l}
{\log _b}(MN) = {\log _b}M + {\log _b}N\\
{\log _b}\left( {\dfrac{M}{N}} \right) = {\log _b}M - {\log _b}N
\end{array}\]