The graphs of y = log[base5](x) and y = log[base5](x+4) are intersected by the line x = k

The distance between the points of intersection is 0.5:
If k = a + b^(1/2) , where a and b are integers, find the value of a + b.

Please help, thanks.

let P & Q be the points of intersections as:
x = k and y = log5(x) => P(k , log5(k))
x = k and y = log5(x+4) => Q(k, log5(k+4))
PQ = log5(k+4) - log5(k)
PQ = log5[(k+4)/k]=> distance of PQ = 0.5:
log5[(k+4)/k = 0.5
(k + 4)/k = 5^(0.5)
1 + 4/k = 5^(1/2)
4/k = 5^(1/2) - 1
1/k = [5^(1/2) - 1]/4
k = 4 /[(5^(1/2) - 1)] * [(5^(1/2) + 1)] / [(5^(1/2) + 1)]
k = 4 * [(5^(1/2) + 1)] / (5 - 1)
K = 1 + 5^(1/2) => k = a + b^(1/2)
a = 1 , b = 5
hence: a + b = 6

Edit: here is a graph for you:
http://www.wolframalpha.com/input/?i=plo…

Merry Christmas.

log[base5](k+4) - log[base5](k) = 0.5 = log[base5] sqrt(5)
(k+4)/k = sqrt(5)
k+4 = ksqrt(5)
k = 4/[sqrt(5) - 1] = 1 + sqrt(5)
So, a = 1, b = 5
a+b = 6 <==Answer