Chapter 4.5, Problem 17E

To determine

To calculate: The solution of the exponential , rounded to four decimal places , $5^{\frac{-x}{100}}=2$

Answer to Problem 17E

The four decimal places rounded value of exponential expression,$x\approx -43.06765$

Explanation of Solution

Given information:The given expression is as, $5^{\frac{-x}{100}}=2$

Formula used:

For the exponential expression take log on both sides.

These are the steps to do this ,

  $In{a}^x=xIna,In{e}^x=xIne$

Step 1 take “In” both side.

Step 2. Then to find the value of x one side take all the values of x and another side all the values of “In” and constants .

Step 3. Then with the help of “In” table put the value of “In” .

Step 4. Take the four decimal value of x from there .

Natural logarithms is “In”

Calculation :

Consider the expression, $5^{\frac{-x}{100}}=2$

Take natural logarithms both sides:

  $\begin{array}{l}{5}^{\frac{-x}{100}}=2\\ In{5}^{\frac{-x}{100}}=In2\end{array}$

Apply the identity: $In{a}^x=xIna$

  $\begin{array}{c}{5}^{\frac{-x}{100}}=2\\ \frac{-x}{100}In\left(5\right)=In2\\ x=-100\frac{In2}{In5}\end{array}$

Apply the value of In2,In5:

Rewrite the expression: $\begin{array}{l}x=-100\frac{In2}{In5}\\ x\approx -43.0676\end{array}$

The above expression can’t be simplified further.

Thus the simplified form of the expression is,$x\approx -43.0676$