Solve \[{{\log }_{3\sqrt{2}}}5832\]

Hint: In this question we have to solve the \[{{\log }_{3\sqrt{2}}}5832\] hence we can change the term in exponential form and mark it as equation one, then we will find out the factors of \[5832\] and then put the factors in the equation so that we can compare the terms to obtain the given result.

Complete step by step answer:
Logarithm is of two types that is common logarithm and natural logarithm. Where the base \[10\] logarithms are also known as the common logarithm. It is written as log \[10\] or just log and the base \[e\] logarithm is the natural logarithm. The natural logarithm is denoted by the letters ln or log \[e\].
Now according to the question we need to find the value of \[{{\log }_{3\sqrt{2}}}5832\]
Let \[{{\log }_{3\sqrt{2}}}5832=x\]
It is given in the form of \[{{\log }_{e}}a=M\] and we know that it is equal to \[{{e}^{M}}=a\]
\[\Rightarrow {{\left( 3\sqrt{2} \right)}^{x}}=5832\] mark it as equation \[(1)\]
As we know that the factor of \[5832\] are:
\[\Rightarrow 5832={{2}^{3}}\times {{3}^{6}}\]
\[\Rightarrow 5832={{\left( \sqrt{2} \right)}^{6}}\times {{3}^{6}}\]
\[\Rightarrow 5832={{\left( 3\sqrt{2} \right)}^{6}}\]
Put the value in equation \[(1)\] we get:
\[\Rightarrow {{\left( 3\sqrt{2} \right)}^{x}}={{\left( 3\sqrt{2} \right)}^{6}}\]
On comparing both sides we will get:
\[\Rightarrow x=6\]

Note:
 People utilized logarithm tables in books to multiply and divide before calculators. A slide rule, an instrument with logarithms written on it, has the same information as a logarithm table. Adding logarithms is the same as multiplying and subtracting logarithms is the same as dividing, logarithms can make multiplication and division of huge numbers easier.