Question

If $x^{\ln \left(\frac{y}{z}\right)}.y^{\ln (xz{)}^2}.z^{\ln \left(\frac{x}{y}\right)}=4^{4lny}$ for any $x>1,y>1$ and $z>1$, then which one of the following is correct?

• A. $\ln y$ is the GM of $\ln x,\ln x,\ln x$ and $\ln z$.
• B. $\ln y$ is the AM of $\ln x,\ln x,\ln x$ and $\ln z$.
• C. $\ln y$ is the HM of $\ln x,\ln x,\ln x$ and $\ln z$.
• D. $\ln y$ is the AM of $\ln x,\ln x,\ln z$ and $\ln z$.

Answer 1

The correct option is B $\ln y$ is the AM of $\ln x,\ln x,\ln x$ and $\ln z$.

Taking natural logarithm on both sides, we have

$\left[\ln \frac{y}{z}\ln x\right]+2\ln \left(xz\right)\ln y+\left[\ln \frac{x}{y}\ln z\right]=4\left[\ln y\right]\ln y$

$⟹[\ln y-\ln z]\ln x+2(\ln x+\ln z)\ln y+[\ln x-\ln y]\ln z=4[\ln y{]}^2$

$⟹3\ln x+\ln z=4\ln y$

$⟹\frac{3\ln x+\ln z}{4}=\ln y$

$⟹\ln y$ is the AM of $\ln x,\ln x,\ln x$ and $\ln z$