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How do you solve$\ln x - \ln 3 = 2$?

Last updated date: 16th Jun 2024
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Hint: An exponent that is written in a special way is known as a logarithm. Logarithm functions are just opposite or inverse of exponential functions. We can easily express any exponential function in a logarithm form. Similarly, all the logarithm functions can be easily rewritten in exponential form. In order to solve this equation, we have to use some of the logarithm function properties.

Complete step by step solution:
Here, in this question we have to solve $\ln x - \ln 3 = 2$ for the value of $x$. This question deals with logarithm functions, which are just the inverse of exponential functions. In order to solve this question, we will have to make use of logarithm function properties.Given is, $\ln x - \ln 3 = 2$.

We know that one of the logarithm function properties is, logarithm quotient rule. The logarithm quotient rule says that if ${\log _b}\left( {\dfrac{x}{y}} \right) = {\log _b}\left( x \right) - {\log _b}\left( y \right)$. By making use of the same property in the given equation we get,
\ln x - \ln 3 = 2 \\
\Rightarrow \ln \left( {\dfrac{x}{3}} \right) = 2 \\
Same as, ${\ln _e}\left( {\dfrac{x}{3}} \right) = 2$.
Now, we take exponential on both the sides of the equation and we get,
\Rightarrow \dfrac{x}{3} = {e^2} \\
\therefore x = 3{e^2} \\ $
Hence, the value of $x$ in $\ln x - \ln 3 = 2$ is $3{e^2}$.

Note: This problem and similar to these can very easily be solved by making use of different logarithm properties. Students should keep in mind the properties of logarithmic functions. Logarithms are useful when we want to work with large numbers. Logarithm has many uses in real life, such as in electronics, acoustics, earthquake analysis and population prediction. When the base of common logarithm is $10$ then, the base of a natural logarithm is number $e$.