The equation 6^{x}+8^{x}=10^{x} has | vedclass

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Question

(C) Given equation: $6^{x}+8^{x}=10^{x}$Divide both sides by $10^{x}$:$\left(\frac{6}{10}\right)^{x}+\left(\frac{8}{10}\right)^{x}=1$$\left(\frac{3}{5

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exactly one real root

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(C) Given equation: $6^{x}+8^{x}=10^{x}$Divide both sides by $10^{x}$:$\left(\frac{6}{10}\right)^{x}+\left(\frac{8}{10}\right)^{x}=1$$\left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}=1$Let $f(x) = \left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}$.Since $\frac{3}{5} Therefore, their sum $f(x)$ is also a strictly decreasing function.$A$ strictly decreasing function can take the value $1$ at most once.By observation, for $x=2$, $\left(\frac{3}{5}\right)^{2}+\left(\frac{4}{5}\right)^{2} = \frac{9}{25}+\frac{16}{25} = \frac{25}{25} = 1$.Thus, $x=2$ is the unique solution.Hence, the equation has exactly one real root.

The equation $6^{x}+8^{x}=10^{x}$ has

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