Question

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)

Answer 1

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