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prove that the equation $$2^x + 3^x + 4^x - 5^x =0$$

has just one root.

ATTEMPT: Write $2^x + 3^x + 4^x = 5^x$. By sketching the graphs it is confirmed that they will intersect at somewhere between $2$ & $3$. That's the only point in the first quadrant.

but can we prove that they will not intersect in the second quadrant?

Besides this can there be a more mathematical approach?

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

LHS is strictly decreasing, continuous for all $x\in\Bbb R$, since $(2/5)^x, (3/5)^x, (4/5)^x$ are. RHS is constant.

$\lim_{x\to -\infty}\text{LHS}=+\infty$ and $\lim_{x\to +\infty}\text{LHS}=0$, so LHS crosses $1$. Exactly one solution exists.

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