Solve the following equations.
$5^{2 x} = 3^{2 x} + {2.5}^{x} + {2.3}^{x}$

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Solution

${5^{x}}^{2} = {3^{x}}^{2} + 2 \times (5^{x} + 3^{x})$
${5^{x}}^{2} - {3^{x}}^{2} = 2 \times (5^{x} + 3^{x})$
$(5^{x} + 3^{x}) \times (5^{x} - 3^{x}) = 2 \times (5^{x} + 3^{x})$
Since $5^{x} + 3^{x}$ not equals to $0$
Hence
$5^{x} - 3^{x} = 2$
only if $x = 1$
So $x = 1$ is only solution of the given equation $.$

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