Chapter 13, Problem 1RE

Let $f(x,y)=e^x\ln y.\text{Find}$

$\begin{array}{l}\text{(a)}f(\ln y,e^x)\\ (\text{}\text{b)}f(r+s,rs).\end{array}$

To determine

To calculate: The value of $f\left(\ln y,e^x\right)$ where function is $f\left(x,y\right)=e^x\ln y$ .

Answer to Problem 1RE

The value of $f\left(\ln y,e^x\right)$ is $xy$ .

Explanation of Solution

Calculation:

Consider the provided function $f\left(x,y\right)=e^x\ln y$ .

Substitute $x=\ln y$ and $y=e^x$ in the provided function,

$f\left(\ln y,e^x\right)=e^{\left(\ln y\right)}\ln \left(e^x\right)$

Use the property of natural logarithm, $\ln \left(a^x\right)=x\ln a$ , $\ln e=1$ , and $e^{\ln a}=a$

$\begin{array}{c}f\left(\ln y,e^x\right)=y\left(1\right)x\ln \left(e\right)\\ =xy\left(1\right)\\ =xy\end{array}$

Hence, the value of $f\left(\ln y,e^x\right)$ is $xy$ .

To determine

To calculate: The value of $f\left(r+s,rs\right)$ where function is $f\left(x,y\right)=e^x\ln y$ .

Answer to Problem 1RE

The value of $f\left(\ln y,e^x\right)$ is $e^{\left(r+s\right)}\ln \left(rs\right)$ .

Explanation of Solution

Calculation:

Consider the provided function $f\left(x,y\right)=e^x\ln y$ .

Substitute $x=r+s$ and $y=rs$ in the provided function,

$f\left(r+s,rs\right)=e^{\left(r+s\right)}\ln \left(rs\right)$

Hence, the value of $f\left(r+s,rs\right)$ is $e^{\left(r+s\right)}\ln \left(rs\right)$ .