Find the equation of a logarithmic function whose graph has x-intercept 8, a vertical asymptote at x=0, and which passes through the point (1/2,4)
Since I see numbers 1/2 , 4, and 8 I will use
log2 as my logarithm
Since the y-axis is an asymptote, there cannot be a horizontal shift of the standard log function, but there could be a vertical shift as well as a compression.
let the function by
y = alog2x + k
point (8,0) is on it, so
0 = alog28 + k
0 = a(3) + k , (#1)
point (1/2,4 lies on it, so
4 = alog2(1/2) + k
4 = a(-1) + k , (#2)
subtract #1 from #2
4 = -4a
a = -1
then 0 = -3+k
k = 3
so one possible log function would be
y = - log2x + 3
To find the equation of a logarithmic function, we need to consider its general form:
y = alog(b(x-h))+k,
where:
a is the vertical stretch/compression factor,
b is the base of the logarithm,
(h, k) represents the coordinates of the vertex or the translation of the logarithmic graph.
Given information:
x-intercept: 8
Vertical asymptote: x = 0
Point: (1/2, 4)
Step 1: Determine the base of the logarithm.
Since there is a vertical asymptote at x = 0, we can conclude that the base of the logarithm is 10 (common logarithm) or e (natural logarithm).
Step 2: Calculate the vertical stretch/compression factor, a.
Since the graph passes through the point (1/2, 4), we can substitute these values into the equation to find a:
4 = alog(b(1/2-0))+k
Since the base of the logarithm can be 10 or e, we have two possible equations:
Case 1: 4 = alog₁₀(1/2)+k
Case 2: 4 = alogₑ(1/2)+k
Step 3: Determine the translation factor, (h, k).
Since the vertical asymptote is at x = 0, the translation factor is (0, k). Therefore, h = 0.
Step 4: Solve for the equation of the logarithmic function.
Let's solve both cases and see which one satisfies all the given information.
Case 1: 4 = alog₁₀(1/2)+k
Substitute the coordinates of the point (1/2, 4):
4 = alog₁₀(1/2)+k
Case 2: 4 = alogₑ(1/2)+k
Substitute the coordinates of the point (1/2, 4):
4 = alogₑ(1/2)+k
Now, we need to solve the two equations simultaneously to find the values of a and k.
To find the equation of a logarithmic function, we can follow these steps:
Step 1: Identify the basic form of a logarithmic function.
The basic form of a logarithmic function is given by:
y = log base a (x)
Step 2: Determine the base of the logarithmic function.
To determine the base, we need to examine the given information.
Given that the logarithmic function has a vertical asymptote at x = 0, it means that the base of the logarithmic function must be greater than 1. This is because logarithmic functions with bases greater than 1 have graphs that approach the vertical asymptote as x approaches negative infinity.
Step 3: Write the general equation of the logarithmic function.
Now that we have the base of the logarithmic function, we can write the general equation as:
y = log base a (x)
Step 4: Use the given information to determine the specific equation.
The x-intercept at 8 means that when x = 8, the value of y will be 0 since the graph crosses the x-axis. Substituting these values into the general equation, we get:
0 = log base a (8)
To solve for the base, we need to rewrite the equation in exponential form:
8 = a^0
Since any number raised to the power of 0 is equal to 1, we can conclude that a = 8.
Therefore, the specific equation becomes:
y = log base 8 (x)
Finally, we need to find the vertical shift using the given point (1/2, 4). We can substitute these values into the equation and solve for the vertical shift (k):
4 = log base 8 (1/2) + k
To solve for k, we isolate it on one side:
4 - log base 8 (1/2) = k
After calculating the value of k, we can write the final equation of the logarithmic function.