Question: Tutorial Exercise Solve the equation for \( x \). \[ e^{-1/x} = e^{1/3} \] Step 1…

Tutorial Exercise

Solve the equation for \( x \).

\[ e^{-1/x} = e^{1/3} \]

Step 1

Note that if \( e^a = e^b \) then

\[ a = \boxed{-3} \]

Solution

To solve the equation \( e^{-1/x} = e^{1/3} \), we follow these steps: Since the bases of the exponents are the same (both are \( e \)), we can set the exponents equal to each other: \[ - \frac{1}{x} = \frac{1}{3} \] Now, solve for \( x \). First, multiply both sides by \(-1\): \[ \frac{1}{x} = -\frac{1}{3} \] Take the reciprocal of both sides to solve for \( x \): \[ x = -3 \] So, the solution is \( x = -3 \).