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How do you solve \[\ln \left( {{x}^{2}} \right)=4\]?

seo-qna
Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint:In the given question, we have been asked to find the value of ‘x’ and it is given that \[\ln \left( {{x}^{2}} \right)=4\]. In order to solve the question, first we need to use the basic property of logarithms i.e. \[\ln \left( {{a}^{b}} \right)=b\ln \left( x \right)\] and \[{{\log }_{b}}\left( x \right)=y\] is equivalent to \[{{b}^{y}}=x\]. Then we simplify the equation further to get the possible values of ‘x’. After applying the properties of logarithm, we will solve the equation in a way we solve general linear equations. Then, we will get the required solution.

Formula used:
\[\ln \left( {{a}^{b}} \right)=b\ln \left( x \right)\]
If \[x\] and b are positive real numbers and b is not equal to 1,
Then \[{{\log }_{b}}\left( x \right)=y\]is equivalent to\[{{b}^{y}}=x\].

Complete step by step solution:
We have given that,
\[\ln \left( {{x}^{2}} \right)=4\]
As, we know that
\[\ln \left( {{a}^{b}} \right)=b\ln a\]
Applying \[\ln \left( {{a}^{b}} \right)=b\ln a\] in the given question, we get
\[\Rightarrow 2\ln \left( x \right)=4\]
Multiplying both the sides of the equation by 2, we get
\[\Rightarrow \dfrac{2\ln \left( x \right)}{2}=\dfrac{4}{2}\]
Simplifying the above equation, we get
\[\Rightarrow \ln \left( x \right)=2\]
Using the definition of log,
If \[x\] and b are positive real numbers and b is not equal to 1,
Then \[{{\log }_{b}}\left( x \right)=y\]is equivalent to \[{{b}^{y}}=x\].
Applying the above property, we get
\[\Rightarrow x={{e}^{2}}\]
By using the calculator,
\[\Rightarrow {{e}^{2}}=7.389\]
Therefore, the possible value of ‘x’ is \[{{e}^{2}}\] or 7.389.
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 be required 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.