Chapter 3.4, Problem 39E

To determine

To find : solution to the exponential equation algebraically.

Answer to Problem 39E

The solution to the equation $4^x=5^{x^2}$ is $\underset{\_}{\boxed{x=0\&x\approx 0.861}}$

Explanation of Solution

Given information: $4^x=5^{x^2}$

Concept Involved:

The word “solve” means the process of finding the value of x that makes the equation true. In order to solve we need to undo whatever is done to x .

Formula Used

  $\ln m^n=n\left(\ln m\right)$

  $\ln e=1$

Calculation:

Take natural logarithm on both sides of the equation

  $\ln 4^x=\ln 5^{x^2}$

Use the logarithmic property $\ln m^n=n\left(\ln m\right)$ on both sides of the equation to rewrite it

  $x\cdot \ln 4=x^2\cdot \ln 5$

Subtracting $x\cdot \ln 4$ on both sides of the equation in order to get x alone in one side of the equation

  $x\cdot \ln 4-x\cdot \ln 4=x^2\cdot \ln 5-x\cdot \ln 4$

Simplify on both sides of the equation

  $0=x^2\cdot \ln 5-x\cdot \ln 4$

Use symmetric property of equality to rewrite the equation

  $x^2\cdot \ln 5-x\cdot \ln 4=0$

Factor x in left side of the equation

  $x\left(x\cdot \ln 5-\ln 4\right)=0$

Use the zero factor property which states that “if $ab=0$ , then $a=0$ or $b=0$ ” to set each factor equal to 0

  $x=0$ ►1^st equation

  $x\cdot \ln 5-\ln 4=0$ ►2^nd equation

Solve the 2^nd equation $x\cdot \ln 5-\ln 4=0$

• By adding ln4 on both sides, then
• By Simplifying on both sides of the equation, then
• By dividing ln5 on both sides, and then
• By Simplifying fraction on both sides

  $x\cdot \ln 5-\ln 4+\ln 4=0+\ln 4$

  $x\cdot \ln 5=\ln 4$

  $\frac{x\cdot \ln 5}{\ln 5}=\frac{\ln 4}{\ln 5}$

  $x=\frac{\ln 4}{\ln 5}$

  $x\approx 0.861$

Check for extraneous solution by substituting the result in the original equation.

Both $x=0\&x\approx 0.861$ makes the original equation TRUE

Conclusion:

The solution to the equation $4^x=5^{x^2}$ is $\underset{\_}{\boxed{x=0\&x\approx 0.861}}$