## PROBLEM 1

Solve in $R$ the equation:

$$
54^{x}+27^{x}+9^{x}+3^{x}=2^{x}
$$
## Solution:

Divide the equation by $2^{\mathrm{x}}$ and we get $27^{x}+\left(\frac{27}{2}\right)^{x}+\left(\frac{9}{2}\right)^{x}+\left(\frac{3}{2}\right)^{x}=1$.

Let the function $f(x)=27^{x}+\left(\frac{27}{2}\right)^{x}+\left(\frac{9}{2}\right)^{x}+\left(\frac{3}{2}\right)^{x}-1$. $3 p$

The function is increasing on $R$, hence $f(x)=0$ admits at most one solution $2 p$

We observe that $x=-1$ is a solution, thus $x=-1$ is the unique solution. $2 p$