[fishsauce]

[ QUOTE ]
c is an arbitrary constant, a is a real number.

ln(a)
------- = c
(1 - a)

Isolate a.

[/ QUOTE ]

The only equivalence I can find with only one occurence of a is

c = \int_1^a (x+1)/x dx

where \int_1^a is the definite integral from 1 to a. This is derived from the original equation by first multplying bth sides by (1-a), then exponentiating both sides to get
a e^a = e^(c+1).
Then taking the natural log of both sides you get
c+1 = ln(a e^a) = a + ln(a).
Since ln(a) = \int_1^a 1/x dx and a = \int_1^a 1 dx + 1,
we have
c+1 = \int_1^a (1 + 1/x) dx + 1.
But I think this is all you can do.

Also, from the second FTC, since c is constant,
0 = (a+1)/a
for which the only solution is a = -1, for which the original problem statement is undefined.

Anyhow, where does this problem come from?