5 Solve the exponential equation

$$
2^{x} \cdot 3^{5^{-x}}+\frac{3^{5^{x}}}{2^{x}}=6
$$
Solution. Applying the arithmetic and geometric means inequality and, subsequently, one of its most well-known consequences (the sum of a positive real number and its inverse is always greater than or equal to 2, and equality only holds for the number 1), we have,

$$
6=2^{x} 3^{5^{-x}}+2^{-x} 3^{5^{x}} \geq 2 \sqrt{2^{x} 3^{5^{-x}} 2^{-x} 3^{5 x}}=6 .
$$

And equality will hold when the numbers being averaged are equal.

$$
2^{x} 3^{5^{-x}}=2^{-x} 3^{5^{x}} \Leftrightarrow 2^{2 x}=3^{5^{x}-5^{-x}} \Leftrightarrow 5^{x}=5^{-x}
$$

which is, when $x=0$, which will be, therefore, the only solution to the equation.