Question

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

Answer 1

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)`