Chapter 8.1, Problem 3E

To determine

The value of x.

Answer to Problem 3E

The value of x is $x=1.765$ .

Explanation of Solution

Given:

  $5^{x-1}=2^x$

Calculation:

  $5^{x-1}=2^x$

Taking logarithm on both sides,

  $\begin{array}{l}{5}^{x-1}=2^x\\ \ln \left(5^{x-1}\right)=\ln \left(2^x\right)\\ \left(x-1\right)\ln 5=x\ln 2\\ x\ln 5-\ln 5=x\ln 2\\ x\ln 5-x\ln 2=\ln 5\\ x\left(\ln 5-\ln 2\right)=\ln 5\\ x=\frac{\ln 5}{\ln 5-\ln 2}\\ x=\frac{1.609}{1.609-0.693}\\ x=\frac{1.609}{0.916}\\ x=1.765\end{array}$

Therefore,

The value of x is $x=1.765$ 2.