Question

How do you expand the logarithmic expression $\ln \left( \dfrac{x}{3}y \right)$?

Answer 1

For the question we are asked to find the logarithmic expansion of $\ln \left( \dfrac{x}{3}y \right)$. So, for the questions of these kind we will use the basic logarithmic formulae which are $\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$and $\ln \left( ab \right)=\ln a+\ln b$. Using the above mentioned logarithmic formulae we will simplify the question and get the solution for the required question.

Complete step by step solution:
Firstly, for the question $\ln \left( \dfrac{x}{3}y \right)$ we will use the basic logarithmic formula which is $\ln \left( ab \right)=\ln a+\ln b$.
After using the formula we will simplify the equation. So, the equation will be reduced as follows.
$\Rightarrow \ln \left( \dfrac{x}{3}y \right)$
$\Rightarrow \ln \left( \dfrac{x}{3} \right)+\ln \left( y \right)$
Here after getting the above equation for the further simplification we will use the formulae $\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$ to the first term in the equation and we will keep the other term as it is.
So, after using the logarithmic formula $\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$ to the first term in the above equation we get the equation reduced as follows.
$\Rightarrow \ln \left( \dfrac{x}{3} \right)+\ln \left( y \right)$
$\Rightarrow \ln \left( x \right)-\ln \left( 3 \right)+\ln y$
Now, we will rearrange the equation which we got after all simplifications in the above to get the solution to look in a more familiar or an easier way.
So, after rearranging the equation will become as follows.
$\Rightarrow \ln \left( x \right)+\ln y-\ln \left( 3 \right)$
Therefore, the solution to the given question will be $\ln \left( x \right)+\ln y-\ln \left( 3 \right)$.

Note: We must be very careful in performing the calculations. We must have a very good knowledge in the concept of logarithms. We must know basic formulae like,
$\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$ and $\Rightarrow \ln \left( ab \right)=\ln a+\ln b$. We must not do mistake in using the formula for example for $\ln \left( ab \right)$ if we use $\ln a-\ln b$ as formula we get $\ln \left( \dfrac{xy}{3} \right)=\ln \dfrac{x}{3}-\ln y$ which makes our whole solution wrong.