I've got an equation where variable I'm looking for is an exponent of $2$ different bases:
$$a^x+b^x=c$$
If I know $a, b$ and $c$, how can I calculate $x$?
I was thinking of taking logarithm of both sides to somehow use the rule that $\log(a^x) = x\cdot\log(a)$, but this doesn't help me much as I have the logarithm of sum on the left side:
$$\log(a^x+b^x)=\log(c)$$
Now I'm a bit stuck at this point, any hints as to how to approach this?
Best Answer
We assume that $1\lt a \lt b.$ Thus, $r:={\log b\over\log a}\gt 1.$ Let $t=a^x.$ Thus, $t^r=b^x.$ Therefore, $a^x+b^x=c$ becomes $t+t^r=c\tag 1$ We assume that $x$ is positive so that $t\gt 1$ and $c\gt 2.$ The derivative of $f(t),$ the left side of (1), is positive for $t>1.$ Since $f(1)\lt c,$ the solution of (1) exists and is unique. Thus, we know $a^x,$ and can calculate $x.$ (For example if $a=2$ and $b=3,$ then $r$ is approximately 1.58496250072 )