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; submit the graph via the Dropbox. State the equation of the line asymptotic to the graph y = log2 x
Do you mean log(base2)of x?
log2(0.001) = log10(.001)/log10(2) = -3/.30103 = -9.97
log2(0.01) = -6.64
log2(0.1) = -3.32
log2(1) = 0
You do the graphing and submitting.
As x-> 0, log2(x) -> -infinity
The negative y axis is the asymptote.
To evaluate the logarithmic equation for different values of x, we will use the equation y = log2(x). Here are the steps to evaluate it for the given values and to plot the graph:
For x values greater than 1:
1) Substitute x = 2 into the equation:
y = log2(2) = 1
So, the first ordered pair is (2, 1).
2) Substitute x = 3 into the equation:
y = log2(3)
Since 3 is not a power of 2, we can use the change of base formula to convert it to base 2:
y = log2(3) = log(3) / log(2)
≈ 1.58496
So, the second ordered pair is (3, 1.58496).
3) Substitute x = 4 into the equation:
y = log2(4) = 2
The third ordered pair is (4, 2).
For x values between 0 and 1:
1) Substitute x = 1/2 into the equation:
y = log2(1/2) = log2(2^-1) = -1
The fourth ordered pair is (1/2, -1).
2) Substitute x = 1/4 into the equation:
y = log2(1/4) = log2(2^-2) = -2
The fifth ordered pair is (1/4, -2).
3) Substitute x = 1/8 into the equation:
y = log2(1/8) = log2(2^-3) = -3
The sixth ordered pair is (1/8, -3).
For x = 1:
1) Substitute x = 1 into the equation:
y = log2(1) = 0
The seventh ordered pair is (1, 0).
To plot the graph, plot the seven ordered pairs on the x-y plane. The x-values range from 1/8 to 4. Connect the points with a smooth curve.
The line asymptotic to the graph y = log2(x) is the vertical line x = 0. This is because log2(x) is undefined for x ≤ 0.