Chapter 4.5, Problem 4E

To determine

To calculate: The solution of the exponential , rounded to four decimal places , ${10}^{-x}=4$

Answer to Problem 4E

The four decimal places rounded value of exponential expression,

  $x=-0.6020$

Explanation of Solution

Given information:

The given expression is as, ${10}^{-x}=4$

Formula used:

For the exponential expression take log on both sides.

These are the steps to do this ,

  $\log a^x=x\log a$

Step 1 take log 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 log and constants .

Step 3. Then with the help of log table put the value of log .

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

Calculation :

Consider the expression, ${10}^{-x}=4$

Take logarithms both sides.

  $\begin{array}{l}\log {10}^x=25\\ x\log 10=25\end{array}$

Apply the identity: $\log a^x=x\log a$

  $\begin{array}{c}\log {10}^{-x}=\log 4\\ -x\log 10=\log 4\\ x=-\frac{\log 4}{\log 10}\end{array}$

Apply the value of log25 and log10:

Rewrite the expression: $\begin{array}{l}x=-\frac{\log 4}{\log 10}\\ x=-0.6020\end{array}$

The above expression can’t be simplified further.

Thus the simplified form of the expression is $x=0.6020$ .