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; submit the graphs via the Dropbox. y=5^(x-2),x=5^(y-2)
To evaluate the exponential equation y = 5^(x-2) for various values of x, we can substitute those values into the equation and calculate the corresponding values of y.
Let's start by evaluating the equation for three positive values of x: x = 1, x = 2, and x = 3.
For x = 1:
y = 5^(1-2) = 5^(-1) = 1/5
For x = 2:
y = 5^(2-2) = 5^0 = 1
For x = 3:
y = 5^(3-2) = 5^1 = 5
So the ordered pairs for the exponential equation y = 5^(x-2) when x is positive are (1, 1/5), (2, 1), and (3, 5).
Next, let's evaluate the equation for three negative values of x: x = -1, x = -2, and x = -3.
For x = -1:
y = 5^(-1-2) = 5^(-3) = 1/125
For x = -2:
y = 5^(-2-2) = 5^(-4) = 1/625
For x = -3:
y = 5^(-3-2) = 5^(-5) = 1/3125
So the ordered pairs for the exponential equation y = 5^(x-2) when x is negative are (-1, 1/125), (-2, 1/625), and (-3, 1/3125).
Now, let's evaluate the equation for x = 0:
y = 5^(0-2) = 5^(-2) = 1/25
The ordered pair for the exponential equation y = 5^(x-2) when x = 0 is (0, 1/25).
To transform the second expression x = 5^(y-2) into the equivalent logarithmic equation, we need to switch the roles of x and y. The logarithmic equation is:
log5(x) = y - 2
Now, let's evaluate the logarithmic equation for three values of x that are greater than 1: x = 10, x = 100, and x = 1000.
For x = 10:
log5(10) = y - 2
y = log5(10) + 2
For x = 100:
log5(100) = y - 2
y = log5(100) + 2
For x = 1000:
log5(1000) = y - 2
y = log5(1000) + 2
We can use a calculator to get the numerical values of y for these equations.
Similarly, let's evaluate the logarithmic equation for three values of x that are between 0 and 1: x = 0.2, x = 0.3, and x = 0.4.
For x = 0.2:
log5(0.2) = y - 2
y = log5(0.2) + 2
For x = 0.3:
log5(0.3) = y - 2
y = log5(0.3) + 2
For x = 0.4:
log5(0.4) = y - 2
y = log5(0.4) + 2
Finally, let's evaluate the logarithmic equation for x = 1:
log5(1) = y - 2
y = log5(1) + 2
Once again, we can use a calculator to get the numerical values of y for these equations.
Collecting all the ordered pairs obtained from both the exponential and logarithmic equations, we can plot the graph of each function.