Lower and upper bound for Chebyshev's psi function

In this post, I prove the inequalities $$\begin{equation}\label{eq:1} \log 2 \cdot x - 3 \log x \le \psi(x) \le 2 \log 2 \cdot x + \frac{3}{\log 2} \log^2 x \end{equation}$$ for all real $x \ge 2$. Hereby is $\psi(x) = \sum_{n \le x } \Lambda(n)$ with $\Lambda$ the von Mangoldt function. This is a second post motivated by Chebyshev's paper "Mémoire sur les nombres premiers" from 1852. In this paper, Chebyshev uses a more complex variant of the calculation below to obtain stronger inequalities. Similar inequalities can also be found on page 50 in Murty's book.

How not to prove Bertrand's postulate

This is a first post motivated by Chebyshev's paper "Mémoire sur les nombres premiers" from 1852. In this paper Chebyshev proves Bertrand's postulate that there is always a prime between $a$ and $2a$ for all $a \ge 2$. Chebyshev bases his proof on inequalities for the function $$\begin{equation}\label{HNTPeq:1} T(x) - T\left( \frac{x}{2} \right)- T\left( \frac{x}{3} \right)- T\left( \frac{x}{5} \right)+ T\left( \frac{x}{30} \right) \end{equation}$$ with $$\begin{equation*} T(x) = \sum_{ 1 \le n \le x} \log n \end{equation*}$$ Chebyshev's paper is quite easy to read and can be found at this link. In this post I will follow Chebyshev's reasoning on an easier version of $\eqref{HNTPeq:1}$.