Chapter 4.5, Problem 7E

To determine

To calculate: The solution of the exponential , rounded to four decimal places , $2^{1-x}=3$

Answer to Problem 7E

The four decimal places rounded value of exponential expression,$x=0.5849$

Explanation of Solution

Given information:

The given expression is as, $2^{1-x}=3$

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, $2^{1-x}=3$

Take logarithms both sides.

  $\begin{array}{c}\log 2^{1-x}=\log 3\end{array}$

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

  $\begin{array}{c}1-x\log 2=\log 3\\ 1-x=\frac{\log 3}{\log 2}\\ x=-\frac{\log 3}{\log 2}-1\end{array}$

Apply the value of log3 and log2:

Rewrite the expression: $\begin{array}{l}x=-\frac{\log 3}{\log 2}-1\\ x=0.5849\end{array}$

The above expression can’t be simplified further.

Thus the simplified form of the expression is $x=0.5849$ .