# 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

Its value would depend on

#### 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

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)#