Question

If $3^x+4^x=5^x$ then find the value of $x$

Answer 1

By trial, $x=2$ is a root of the equation . Dividing the equation by $5^x$ we get
${\left(\frac{3}{5}\right)}^x+{\left(\frac{4}{5}\right)}^x=1$
If $x<2,$ then $(3/5{)}^x>(3/5{)}^2$
and $(4/5{)}^x>(4/5{)}^2$ and so on
$(3/5{)}^x+(4/5{)}^x>(3/5{)}^2+(4/5{)}^2=1$
Hence (1) is not satisfied for any value of $x<2$.
Similarly, for $x>2,$ we will always have the inequality $(3/5{)}^x+(4/5{)}^x<1$
Thus $x=2$ is the only root of the given equation. Note that it is immaterial how we found a root, but it is necessary to show that there are no other roots.