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 than 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=2^y-2
To evaluate the exponential equation y = 2^x - 2 for three positive values of x, let's choose x = 1, 2, and 3:
For x = 1:
y = 2^1 - 2
y = 2 - 2
y = 0
For x = 2:
y = 2^2 - 2
y = 4 - 2
y = 2
For x = 3:
y = 2^3 - 2
y = 8 - 2
y = 6
Now, let's evaluate the equation for three negative values of x. We'll choose x = -1, -2, and -3:
For x = -1:
y = 2^-1 - 2
y = 1/2 - 2
y = -1.5
For x = -2:
y = 2^-2 - 2
y = 1/4 - 2
y = -1.75
For x = -3:
y = 2^-3 - 2
y = 1/8 - 2
y = -1.875
Finally, let's evaluate the equation at x = 0:
y = 2^0 - 2
y = 1 - 2
y = -1
Transforming the equation into the equivalent logarithmic equation, we have:
x = 2^y - 2
To evaluate the logarithmic equation for three values of x that are greater than 1, let's choose x = 2, 4, and 8:
For x = 2:
2 = 2^y - 2
4 = 2^y
y = log2(4)
y = 2
For x = 4:
4 = 2^y - 2
6 = 2^y
y = log2(6)
y ≈ 2.585
For x = 8:
8 = 2^y - 2
10 = 2^y
y = log2(10)
y ≈ 3.322
Next, evaluate the logarithmic equation for three values of x that are between 0 and 1. Let's choose x = 0.5, 0.25, and 0.125:
For x = 0.5:
0.5 = 2^y - 2
2.5 = 2^y
y = log2(2.5)
y ≈ 1.3219
For x = 0.25:
0.25 = 2^y - 2
2.25 = 2^y
y = log2(2.25)
y ≈ 1.1699
For x = 0.125:
0.125 = 2^y - 2
2.125 = 2^y
y = log2(2.125)
y ≈ 1.0769
Lastly, evaluate the logarithmic equation at x = 1:
1 = 2^y - 2
3 = 2^y
y = log2(3)
y ≈ 1.585
The resulting ordered pairs for both the exponential and logarithmic equations are as follows:
Exponential equation:
(1, 0), (2, 2), (3, 6), (-1, -1.5), (-2, -1.75), (-3, -1.875), (0, -1)
Logarithmic equation:
(2, 2), (4, 2.585), (8, 3.322), (0.5, 1.3219), (0.25, 1.1699), (0.125, 1.0769), (1, 1.585)
Plotting these ordered pairs on a graph will show the visual representation of each function.
To evaluate the exponential equation for positive values of x, negative values of x, and at x=0, we substitute the given values of x into the equation and calculate the corresponding values of y.
Let's start with positive values of x:
1. Substitute x = 1 into the equation:
y = 2^1 - 2 = 2 - 2 = 0
So, when x=1, y=0. We have the ordered pair (1,0).
2. Substitute x = 2 into the equation:
y = 2^2 - 2 = 4 - 2 = 2
So, when x=2, y=2. We have the ordered pair (2,2).
3. Substitute x = 3 into the equation:
y = 2^3 - 2 = 8 - 2 = 6
So, when x=3, y=6. We have the ordered pair (3,6).
Now let's evaluate for negative values of x:
1. Substitute x = -1 into the equation:
y = 2^-1 - 2 = 1/2 - 2 = -3/2
So, when x=-1, y=-3/2. We have the ordered pair (-1,-3/2).
2. Substitute x = -2 into the equation:
y = 2^-2 - 2 = 1/4 - 2 = -7/4
So, when x=-2, y=-7/4. We have the ordered pair (-2,-7/4).
3. Substitute x = -3 into the equation:
y = 2^-3 - 2 = 1/8 - 2 = -15/8
So, when x=-3, y=-15/8. We have the ordered pair (-3,-15/8).
Next, let's evaluate at x=0:
Substitute x = 0 into the equation:
y = 2^0 - 2 = 1 - 2 = -1
So, when x=0, y=-1. We have the ordered pair (0,-1).
Now, let's transform the second expression into an equivalent logarithmic equation.
The given equation is:
x = 2^y - 2
To transform it into a logarithmic equation, we can rewrite it as follows:
2^y = x + 2
Now, taking the logarithm (base 2) of both sides, we get:
y = log2(x + 2)
Now, we can evaluate the logarithmic equation for different values of x.
For values of x greater than 1:
1. Substitute x = 2 into the logarithmic equation:
y = log2(2 + 2) = log2(4) = 2
So, when x=2, y=2. We have the ordered pair (2,2).
2. Substitute x = 3 into the logarithmic equation:
y = log2(3 + 2) = log2(5) ≈ 2.3219
So, when x=3, y≈2.3219. We have the ordered pair (3,2.3219).
3. Substitute x = 4 into the logarithmic equation:
y = log2(4 + 2) = log2(6) ≈ 2.5849
So, when x=4, y≈2.5849. We have the ordered pair (4,2.5849).
Next, for values of x between 0 and 1:
1. Substitute x = 0.5 into the logarithmic equation:
y = log2(0.5 + 2) = log2(2.5) ≈ 1.3219
So, when x=0.5, y≈1.3219. We have the ordered pair (0.5,1.3219).
2. Substitute x = 0.2 into the logarithmic equation:
y = log2(0.2 + 2) = log2(2.2) ≈ 1.1375
So, when x=0.2, y≈1.1375. We have the ordered pair (0.2,1.1375).
3. Substitute x = 0.8 into the logarithmic equation:
y = log2(0.8 + 2) = log2(2.8) ≈ 1.4854
So, when x=1, y≈1.4854. We have the ordered pair (0.8,1.4854).
Finally, when x = 1:
Substitute x = 1 into the logarithmic equation:
y = log2(1 + 2) = log2(3) ≈ 1.5849
So, when x=1, y≈1.5849. We have the ordered pair (1,1.5849).
To plot the graphs of both functions, we can use the obtained ordered pairs. The exponential function is y=2^x-2 and the logarithmic function is x=2^y-2. Plotting the points will give us a visual representation of the functions on a graph.