Question

Find the no. of solutions of the equation $2^x+3^x+4^x-5^x=0$

Answer 1

Given

$2^x+3^x+4^x=5^x$

$\Rightarrow (\frac{2}{5}{)}^x+(\frac{3}{5}{)}^x+(\frac{4}{5}{)}^x=1$

let

$g\left(x\right)=(\frac{2}{5}{)}^x+(\frac{3}{5}{)}^x+(\frac{4}{5}{)}^x$

clearly g(x) is a strictly decreasing function.

Hence, It is one- one function.

$\therefore$ Number of solution on g(x) is '1'