7) if $\\ln \\frac{\\sqrt{x}}{y^3} = a\\ln x - b\\ln y$, find the value of $a$.
7) if $\\ln \\frac{\\sqrt{x}}{y^3} = a\\ln x - b\\ln y$, find the value of $a$.
Answer
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:
$\frac{1}{2}$