How do you solve\[\ln \left( x \right)-4\ln 3=\ln \left( \dfrac{5}{x} \right)\]?

Hint:In the given question, we have been asked to solve\[\ln \left( x \right)-4\ln 3=\ln \left( \dfrac{5}{x} \right)\]. In order to solve the question for the value of ‘x’, first we apply the property of logarithm which states that \[a\ln b=\ln \left( {{b}^{a}} \right)\]and later by applying the quotient property of logarithmic function we will simplify further. Then to eliminate the log function we need to convert the logarithmic equation into the exponential equation and solve the equation further in a way we solve the general equation.

Formula used:
● The property of logarithm, states that \[a\ln b=\ln \left( {{b}^{a}} \right)\].
● The quotient property of logarithm which states that \[\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)\].

Complete step by step solution:
We have given that,
\[\ln \left( x \right)-4\ln 3=\ln \left( \dfrac{5}{x} \right)\]
Using the property of logarithm, i.e.
\[a\ln b=\ln \left( {{b}^{a}} \right)\]
Applying the property, we get
\[\ln \left( x \right)-\ln \left( {{3}^{4}} \right)=\ln \left( \dfrac{5}{x} \right)\]
Using the quotient property of logarithm, i.e.
\[\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)\]
Applying the quotient property of log, we get
\[\Rightarrow \ln \left( \dfrac{x}{{{3}^{4}}} \right)=\ln \left( \dfrac{5}{x} \right)\]
To eliminate log function or to cancel out the log function, we raise ‘e’ to the power log.
Converting the logarithmic equation into exponential form, we get
\[\Rightarrow {{e}^{\ln \left( \dfrac{x}{{{3}^{4}}} \right)}}={{e}^{\ln \left( \dfrac{5}{x} \right)}}\]
As we know that, \[{{e}^{\ln \left( x \right)}}=x\]
Thus,
\[\Rightarrow \dfrac{x}{{{3}^{4}}}=\dfrac{5}{x}\]
Cross multiplication in the above equation, we get
\[\Rightarrow {{x}^{2}}=5\times {{3}^{4}}\]
Simplifying the above equation, we get
\[\Rightarrow x=\pm \sqrt{5\times {{3}^{4}}}=\pm \sqrt{5}\times {{3}^{2}}\]
Therefore,
\[\Rightarrow x=+\sqrt{5}\times {{3}^{2}}=9\sqrt{5}\] and \[x=-\sqrt{5}\times {{3}^{2}}=-9\sqrt{5}\]
As the log function cannot take negative values, so x = \[-9\sqrt{5}\] is not a solution.
Thus, the possible value of ‘x’ is \[9\sqrt{5}\] .
It is the required solution.

Note: In the given question, we need to find the value of ‘x’. To solve these types of questions, we used the basic formulas of logarithm. Students should always require to keep in mind all the formulae for solving the question easily. After applying log formulae to the equation, we need to solve the equation in the way we solve general linear equations. Students should always remember that natural log and the exponential functions are the inverse of each other, which means that if we raise the exponential function by the natural log of x, then only we would be able to find the value of ‘x’.