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How do you solve \[\ln \left( {{x}^{7}} \right)-\ln \left( {{x}^{2}} \right)=5\]?

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Last updated date: 04th Mar 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 \[\ln \left( {{x}^{7}} \right)-\ln \left( {{x}^{2}} \right)=5\]. In order to find the value of ‘x’, first we will apply the quotient property of logarithm which states that \[{{\log }_{b}}m-{{\log }_{b}}n={{\log }_{b}}\left( \dfrac{m}{n} \right)\] . Then we need to apply the definition of logarithm, which states that \[\log \left( {{x}^{a}} \right)=a\log x\] and then simplify the equation further. After applying log formulae to the equation, we need to solve the equation in the way we solve general linear equations.

Formula used:
● The properties of logarithm to combine two natural logs;
Using the quotient property of logarithm, which states that \[{{\log }_{b}}m-{{\log }_{b}}n={{\log
}_{b}}\left( \dfrac{m}{n} \right)\]
● The definition of logarithm, says that \[\log \left( {{x}^{a}} \right)=a\log x\]

Complete step by step solution:
We have given that,
\[\ln \left( {{x}^{7}} \right)-\ln \left( {{x}^{2}} \right)=5\]
The properties of logarithm to combine two natural logs;
Using the quotient property of logarithm, i.e.
\[{{\log }_{b}}m-{{\log }_{b}}n={{\log }_{b}}\left( \dfrac{m}{n} \right)\]
Applying the property in the above equation, we get
\[\Rightarrow \ln \left( \dfrac{{{x}^{7}}}{{{x}^{2}}} \right)=5\]
On simplifying the above equation, we get
\[\Rightarrow \ln \left( {{x}^{5}} \right)=5\]
By the definition of logarithm, i.e.
\[\log \left( {{x}^{a}} \right)=a\log x\]
Using this, we get
\[\Rightarrow 5\ln \left( x \right)=5\]
Multiplying both the sides of the equation by 5, we get
\[\Rightarrow \dfrac{5\ln \left( x \right)}{5}=\dfrac{5}{5}\]
On simplifying the above, we get
\[\Rightarrow \ln \left( x \right)=1\]
Therefore,
\[\Rightarrow x=e\]
Thus, the value of ‘x’ equals to ‘e’ 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.

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