Evaluate the exponential equation for three positive values of x, three negative values of x, and at x=0. Transform the second expression into the equivalent logarithmic equation; and evaluate the logarithmic equation for three values of x that are greater ththan 1, three values of x that are between 0 and 1, and at x=1. Show your work. Use the resulting ordered pairs to plot the graph of each function.

y=2^x+2, x=4^y+2

To evaluate the exponential equation for three positive values of x, three negative values of x, and at x=0, we substitute the values of x into the equation and calculate the corresponding y values.

Positive values of x:
1. When x = 1: y = 2^(1+2) = 2^3 = 8
2. When x = 2: y = 2^(2+2) = 2^4 = 16
3. When x = 3: y = 2^(3+2) = 2^5 = 32

Negative values of x:
1. When x = -1: y = 2^(-1+2) = 2^1 = 2
2. When x = -2: y = 2^(-2+2) = 2^0 = 1
3. When x = -3: y = 2^(-3+2) = 2^(-1) = 0.5

At x = 0: y = 2^(0+2) = 2^2 = 4

Now, let's transform the second expression into an equivalent logarithmic equation:
x = 4^(y+2)
We can rewrite this as:
4^x = y + 2

To evaluate the logarithmic equation for three values of x that are greater than 1, three values of x that are between 0 and 1, and at x=1, we substitute the values of x into the equation and calculate the corresponding y values.

Values of x greater than 1:
1. When x = 2: 4^2 = y + 2 => 16 = y + 2 => y = 16 - 2 = 14
2. When x = 3: 4^3 = y + 2 => 64 = y + 2 => y = 64 - 2 = 62
3. When x = 4: 4^4 = y + 2 => 256 = y + 2 => y = 256 - 2 = 254

Values of x between 0 and 1:
1. When x = 0.2: 4^0.2 = y + 2 => 1.3195079 = y + 2 => y = 1.3195079 - 2 = -0.68
2. When x = 0.5: 4^0.5 = y + 2 => 2 = y + 2 => y = 2 - 2 = 0
3. When x = 0.8: 4^0.8 = y + 2 => 4.88876 = y + 2 => y = 4.88876 - 2 = 2.88876

At x = 1: 4^1 = y + 2 => 4 = y + 2 => y = 4 - 2 = 2

The resulting ordered pairs for the exponential equation (y=2^x+2) are:
(1, 8), (2, 16), (3, 32), (-1, 2), (-2, 1), (-3, 0.5), (0, 4)

The resulting ordered pairs for the logarithmic equation (4^x = y + 2) are:
(2, 14), (3, 62), (4, 254), (0.2, -0.68), (0.5, 0), (0.8, 2.88876), (1, 2)

Plotting the graph of each function using the ordered pairs will give a visual representation of the exponential and logarithmic equations.