Chapter 3.4, Problem 12E

To determine

To find: The value of the variable x for the equation.

Answer to Problem 12E

The value of the variable x for the equation $e^x=\frac{1}{3}$ is $x=-1.099$ .

Explanation of Solution

Given information:

The equation $e^x=\frac{1}{3}$ .

Formula used:

Inverse property states exponential and logarithmic function are reciprocal to each other and mathematically expressed as $\ln e^x=x$ .

Quotient property of natural logarithmic function is $\ln \frac{x}{y}=\ln x-\ln y$ .

Calculation:

Consider the provided equation, $e^x=\frac{1}{3}$ .

Recall that Inverse property states exponential and logarithmic function are reciprocal to each other and mathematically expressed as $\ln e^x=x$ .

Apply it, take logarithmic function on both sides,

  $\begin{array}{c}\ln e^x=\ln \frac{1}{3}\\ x=\ln \frac{1}{3}\end{array}$

Apply the quotient property of natural logarithmic function that is $\ln \frac{x}{y}=\ln x-\ln y$ .

  $\begin{array}{c}x=\ln 1-\ln 3\\ x=0-\ln 3\\ x=-\ln 3\\ x=-1.097\end{array}$

Since, $\ln 1=0$ .

Hence, the value of the variable x for the equation $e^x=\frac{1}{3}$ is $x=-1.099$ .