Question

1. if $\ln \frac{\sqrt{x}}{y^3} = a\ln x - b\ln y$, find the value of $a$.

Explanation:

Step1: Simplify the left - hand side using logarithm properties

We know that the quotient rule of logarithms is $\ln\frac{m}{n}=\ln m-\ln n$, and the power rule is $\ln a^{b}=b\ln a$.
For $\ln\frac{\sqrt{x}}{y^{5}}$, first, $\sqrt{x}=x^{\frac{1}{2}}$.
Using the quotient rule: $\ln\frac{\sqrt{x}}{y^{5}}=\ln\sqrt{x}-\ln y^{5}$.
Then, using the power rule: $\ln\sqrt{x}=\ln x^{\frac{1}{2}}=\frac{1}{2}\ln x$ and $\ln y^{5} = 5\ln y$.
So, $\ln\frac{\sqrt{x}}{y^{5}}=\frac{1}{2}\ln x-5\ln y$.

Step2: Compare with the given form

The given form is $A\ln x - B\ln y$.
By comparing $\frac{1}{2}\ln x-5\ln y$ with $A\ln x - B\ln y$, we can see that the coefficient of $\ln x$ is $A=\frac{1}{2}$.

Answer:

$\frac{1}{2}$