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 x = 2^y
To evaluate the exponential equation y = 2^x for three positive values of x, three negative values of x, and at x=0:
For positive values of x:
1. Let's choose x = 1, x = 2, and x = 3.
Substitute x = 1: y = 2^1 = 2.
Substitute x = 2: y = 2^2 = 4.
Substitute x = 3: y = 2^3 = 8.
The ordered pairs are (1, 2), (2, 4), (3, 8).
For negative values of x:
1. Let's choose x = -1, x = -2, and x = -3.
Substitute x = -1: y = 2^(-1) = 1/2.
Substitute x = -2: y = 2^(-2) = 1/4.
Substitute x = -3: y = 2^(-3) = 1/8.
The ordered pairs are (-1, 1/2), (-2, 1/4), (-3, 1/8).
At x=0:
1. Substitute x = 0: y = 2^0 = 1.
The ordered pair is (0, 1).
To transform the second expression x = 2^y into the equivalent logarithmic equation:
We can rewrite the equation x = 2^y as y = log2(x).
To evaluate the logarithmic equation y = log2(x) for three values of x > 1, three values of x between 0 and 1, and at x = 1:
For x > 1:
1. Let's choose x = 2, x = 4, and x = 8.
Substitute x = 2: y = log2(2) = 1.
Substitute x = 4: y = log2(4) = 2.
Substitute x = 8: y = log2(8) = 3.
The ordered pairs are (2, 1), (4, 2), (8, 3).
For x between 0 and 1:
1. Let's choose x = 1/2, x = 1/4, and x = 1/8.
Substitute x = 1/2: y = log2(1/2) = -1.
Substitute x = 1/4: y = log2(1/4) = -2.
Substitute x = 1/8: y = log2(1/8) = -3.
The ordered pairs are (1/2, -1), (1/4, -2), (1/8, -3).
At x = 1:
1. Substitute x = 1: y = log2(1) = 0.
The ordered pair is (1, 0).
By plotting the resulting ordered pairs from the exponential and logarithmic equations, we can obtain the graph of each function.
To evaluate the exponential equation y = 2^x, we need to substitute three positive values of x, three negative values of x, and x = 0 into the equation.
Let's begin with positive values of x:
1. For x = 1: y = 2^1 = 2
2. For x = 2: y = 2^2 = 4
3. For x = 3: y = 2^3 = 8
Now let's move on to negative values of x:
1. For x = -1: y = 2^(-1) = 1/2
2. For x = -2: y = 2^(-2) = 1/4
3. For x = -3: y = 2^(-3) = 1/8
Lastly, let's evaluate the exponential equation at x = 0:
For x = 0: y = 2^0 = 1
Now let's transform the second expression x = 2^y into an equivalent logarithmic equation. To do this, we write the logarithmic form of the exponential equation as follows:
x = log(base 2) y
To evaluate the logarithmic equation for three values of x greater than 1, three values of x between 0 and 1, and x = 1, we can use the logarithmic properties. Remember that the logarithm (log) of a base to 1 is always zero.
For values of x greater than 1:
1. For x = 2: log(base 2) 2 = 1
2. For x = 4: log(base 2) 4 = 2
3. For x = 8: log(base 2) 8 = 3
For values of x between 0 and 1:
1. For x = 1/2: log(base 2) (1/2) = -1
2. For x = 1/4: log(base 2) (1/4) = -2
3. For x = 1/8: log(base 2) (1/8) = -3
Lastly, let's evaluate the logarithmic equation at x = 1:
For x = 1: log(base 2) 1 = 0
We now have the following ordered pairs for the exponential equation:
(1, 2), (2, 4), (3, 8), (-1, 1/2), (-2, 1/4), (-3, 1/8), (0, 1)
And the following ordered pairs for the logarithmic equation:
(2, 1), (4, 2), (8, 3), (1/2, -1), (1/4, -2), (1/8, -3), (1, 0)
Using these ordered pairs, we can plot the graph of each function.