y = a log e (x - b) + c
The vertical Asymptotes of the log function are Zeros of its Argument.
Hence (x - b) = 0, Therefore x = b
Comparing with the given equation of Vertical asymptote; x = 1
Therefore we get b = x = 1; Therefore b = 1
So given equation becomes... y = a log e (x -1) + c
Now substitute given point (x, y) as (3, 10)
10 = a log e (3-1) + c
10 = a log e 2 + c .....................(1)
Similarly, the point (5, 12) gives
12 = a log e (5-1) + c
12 = a log e 4 + c .....................(2)
Subtract Eqn (1) from Eqn (2) to eliminate c
2 = a log e 4 - a log e 2
2= a (log e 2^2 - log e 2) = a ( 2 log e 2 - log e 2)
2 = a log e 2
Therefore a = 2 / log e 2
Substitute value of a in Equation (1)
(1) ...... 10 = (2 / log e 2) x (log e 2) + c
10 = 2 + c Therefore c = 8
Therefore a = 2 / Log e 2; b = 1; c = 8