## Task $1 / 84$

Solve the equation $1984^{\lg x}=2 \cdot 1984^{2}-x^{\lg 1984}$ in real numbers $x$.
The given equation is a special case of the equation

$$
a^{\lg x}=2 a^{2}-x^{\lg a}
$$

It holds that

$$
\lg a^{\lg x}=\lg x \cdot \lg a=\lg x^{\lg a}
$$

Thus, due to the uniqueness of the lg function,

$$
a^{\lg x}=x^{\lg a}
$$

Therefore, (1) is equivalent to the equation

$$
a^{\lg x}=2 a^{2}-a^{\lg x}
$$

By further equivalent transformation, we get

$$
2 a^{\lg x}=2 a^{2} \quad ; \quad \lg x=2
$$

and thus $x=100$ as the unique solution independent of the parameter a.