part a: explain why the x-coordinates of the points where the graphs of the equations y = 2^x and y = 4^x−2 intersect are the solutions of the equation 2^x = 4^x−2. (4 points)

part b: make tables to find the solution to 2^x = 4^x−2. take the integer values of x between −4 and 4. (4 points)

part c: how can you solve the equation 2^x = 4^x−2 graphically? (2 points)

A) Take the equation 2^x = 4^x - 2.

Write the equation in exponential form.

2^x = (2²)^x - 2

Using (a^m)^n = (a^n)^m, rewrite the equation.

2^x = (2^x)² - 2

Basically this equation states that 2 to the x = 2 to the x squared minus 2. You can use substitution.

Rewrite the equation saying that x = x² - 2.

Solve for x by using the quadratic formula.

Move x to the other side.

0 = x² - x - 2.

After a series of long steps, you would get :

x = 2 and x = -1.

Knowing this information, substitute the numbers into the previous formula : x = 2^x

2 = 2^x and -1 = 2^x

Solve for x.

1) 2^x = 2

2^x = 2¹

Since the bases are the same, the exponents are equal.

x = 1

2) -1 = 2^x

The bases are different signs so there is no solution.

Ultimately, x = 1.

If you look at the graph above, this answer is supported because the graphs intersect at where x = 1.

The point of intersection between the two graphs is (1,2).

B) If you having graphing calculator, you can input the equations 2^x and 4^x - 2 and go to check the tables for each function.

C) Use a graphing calculator to graph both functions individually and see where both functions intersect. The point on intersection is where 2^x = 4^x - 2.