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)

To find the equation of a logarithmic function with these given characteristics, we can start by considering the general form of a logarithmic function:

y = a * log(base b)(x - h) + k

Where:
- "a" is the vertical stretch or compression factor,
- "b" is the base of the logarithm,
- (h, k) represents the horizontal and vertical shifts, and
- (x, y) are the coordinates on the graph.

Using the given information, we can determine the specific values for these parameters in order to find the equation.

1. x-intercept at 8:
Since the x-intercept occurs when y = 0, we know that (8, 0) is a point on the graph. Plugging these values into the equation:

0 = a * log(base b)(8 - h) + k

2. Vertical asymptote at x=0:
A vertical asymptote occurs when x approaches a specific value, in this case x = 0. This means that as x approaches 0, the logarithmic function approaches negative infinity. Therefore, we can write this condition as:

lim(x->0^+) a * log(base b)(x - h) + k = -∞

3. Passes through the point (1/2, 4):
Plugging these values into the equation, we get:

4 = a * log(base b)((1/2) - h) + k

We now have a system of three equations with three unknowns, "a", "b", and "h":

0 = a * log(base b)(8 - h) + k
lim(x->0^+) a * log(base b)(x - h) + k = -∞
4 = a * log(base b)((1/2) - h) + k

Solving this system of equations will give us the values for "a", "b", and "h", allowing us to write the equation for the logarithmic function based on the given characteristics.