Chapter 11.4, Problem 63SR

To determine

To write an equivalent exponential or logarithmic function.

Answer to Problem 63SR

  $\boxed{x=-\ln 5}$

Explanation of Solution

Given information:

Write an equivalent exponential or logarithmic function.

  $e^{-x}=5$

Calculation:

Here, we will consider an exponential function:

  $e^{-x}=5$

Now, take logarithm to the base $e$ on both sides,

  ${\log }_e\left(e^{-x}\right)={\log }_{e}5$

As, $\log \left(m^n\right)=n\log m$ so the above function becomes,

  $-x{\log }_{e}e={\log }_{e}5$

As, ${\log }_{a}a=1$ so the above function becomes,

  $-x\left(1\right)={\log }_{{}_e}5$

Now, multiply both sides by $-1$ ,

  $x=-{\log }_{e}5$

Since, logarithm function to the base $e$ is natural logarithm,

  $x=-\ln 5$

Hence, the exponential function $e^{-x}=5$ written as logarithmic function becomes $\boxed{x=-\ln 5}$ .