Answer
a.) $x = \frac{1}{e}$
b.) $x = -1$
Work Step by Step
$Use$ $the$ $definition$ $of$ $the$ $logarithmic$ $function$ $to$ $find$ $x:$
a.) $\ln x = -1$
b.) $\ln (\frac{1}{e}) = x$
a.) $\ln x = -1$
Rewrite the natural log form to exponential form: $\ln a = b \rightarrow e^b = a$ [Note: e stands for euler's number, NOT a variable]
$$\ln x = -1 \rightarrow e^{-1} = x$$
$$x = \frac{1}{e}$$
b.) $\ln (\frac{1}{e}) = x$
Rewrite the natural log form to exponential form: $\ln a = b \rightarrow e^b = a$ [Note: e stands for euler's number, NOT a variable]
$$ \ln (\frac{1}{e}) = x \rightarrow e^x = \frac{1}{e}$$
Rewrite $\frac{1}{e}$ to $e^{-1}$
$$e^x = e^{-1}$$
$$x = -1$$