If x^a+x^b = x^c then what is c in terms of a and b?

1 Answer
Apr 14, 2016

There is no expression for c in terms of just a and b.
Its value would depend on x too.

Explanation:

I think this question is inspired by the identities:

x^a * x^b = x^(a+b)

(x^a)^b = x^(ab)

However, when we get to:

x^a+x^b = x^c

there is no simple expression for c in terms of a and b. It will depend on the value of x too.

Taking logs of both sides we get:

log(x^a+x^b) = log(x^c) = c log(x)

So:

c = log(x^a+x^b)/log(x) = log_x(x^a+x^b)

A particular concrete example would be:

2^1 + 2^1 = 4 = 2^2

4^1 + 4^1 = 8 = 4^(3/2)