Q. 70 - Precalculus Enhanced with Graphing Utilities

Question

solve each equation. Verify your results using a graphing utility.

$4^{x^2}=2^x$

Step-by-Step Solution

Answer

The value of x for which the above equation is valid are :

$0 , 1/2$

Step 1. Given equation

$4^{x^2}=2^x$

Step 2. Solving exponentially

$$\begin{gathered} 4^{x^2}=2^x \\ {\left(2^2\right)}^{x^2}=2^x \\ {\left(2\right)}^{2x^2}=2^x \end{gathered}$$

By exponential property 

$$\begin{gathered} a^v=a^u \\ \Rightarrow v=u \end{gathered}$$

Using this :

$$\begin{gathered} 2x^2=x \\ 2x^2-x=0 \\ x(2x-1)=0 \end{gathered}$$

We get values of x :

$0, 1/2$

Step 3. Verifying Graphically

Sketching the graphs of 

$4^{x^2} \& 2^x$

Where they will intersect will be the solution .

they intersect at point 0, 0.5

Hence verified.