Question

Find the value of \[{\log _3}{\log _2}{\log _{\sqrt 3 }}81\].

Answer 1

Hint: Here, we will use the power rule of logarithm, \[{\log _b}\left( {{a^c}} \right) = c{\log _b}a\] and the property of logarithm, \[{\log _b}b = 1\] accordingly in the given expression to simplify it.

Complete step by step solution: We are given that the \[{\log _3}{\log _2}{\log _{\sqrt 3 }}81\].

We can also rewrite the given expression as
\[ \Rightarrow {\log _3}{\log _2}{\log _{\sqrt 3 }}{\left( {\sqrt 3 } \right)^8}\]

Let us now make use of the power rule of logarithm, \[{\log _b}\left( {{a^c}} \right) = c{\log _b}a\].

So, on applying this rule in the above equation, we get

\[ \Rightarrow {\log _3}{\log _2}8{\log _{\sqrt 3 }}\sqrt 3 \]

We will now use the property of logarithm, \[{\log _b}b = 1\] in the above equation, we get

\[
   \Rightarrow {\log _3}{\log _2}8 \cdot 1 \\
   \Rightarrow {\log _3}{\log _2}8 \\
   \Rightarrow {\log _3}{\log _2}{\left( 2 \right)^3} \\
 \]

Using the power rule of logarithm, \[{\log _b}\left( {{a^c}} \right) = c{\log _b}a\] again in the above equation, we get

\[ \Rightarrow {\log _3}3{\log _2}2\]

We can use the property of logarithm, \[{\log _b}b = 1\] again in the above expression to simplify it.

\[
   \Rightarrow {\log _3}3 \cdot 1 \\
   \Rightarrow {\log _3}3 \\
 \]

Applying the rule of the logarithm, \[{\log _b}b = 1\] in the above equation, we get

\[ \Rightarrow 1\]

Thus, the value of \[{\log _3}{\log _2}{\log _{\sqrt 3 }}81\] is 1.

Note: The power rule can be used for fast exponent calculation using multiplication operation. Students should make use of the appropriate formula of logarithms wherever needed and try to solve the problem. One fascinating fact with the log function is that there is no negative value of root in the function.