\documentclass[a4paper,11pt]{report}
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\usepackage{amssymb}
\usepackage[fleqn,leqno]{amsmath}

\DeclareMathOperator{\Log}{\mathrm{Log}}
\renewcommand{\theequation}{\arabic{equation}}

\begin{document}
\chapter{Logaritmi ed Esponenziali}
\section{Proprietà fondamentali}
\paragraph{Esponenziali}
\begin{align}
a^x\cdot a^y&=a^{x+y}\\
\frac{a^x}{a^y}&=a^{x-y}\\
(a^x)^y&=a^{x\cdot \,y}\\
a^x \cdot b^x &=(ab)^x
\end{align}

\paragraph{Logaritmi}
\begin{align}
\log a + \log b &= \log ab\\
\log a - \log b &= \log \frac{a}{b}\\
b\log a &= \log a^b  \intertext{ in particolare per $b=-1$ si ha} 
-\log a &= \log \frac{1}{a} \\
\log_a b &= \frac{\log_c b}{\log_c a} \intertext{ in particolare per $c=b$ si ha}
\log_a b&= \frac{1}{\log_b a}\\
\log_{a} b&= \log_{a^c} b^c
\end{align}

\section{Esercizi}
\begin{align}
&x+\Log(1+2^x)=x\Log 5+\Log 6\\
&4x^2+3^{\sqrt{x}+1}+x\cdot3^{\sqrt{x}}<2x^2\cdot3^{\sqrt{x}}+2x+6\\
&A=\{z\in\mathbb{C}\;|\;z^8=1;|\Re (z)|=|\Im(z)|;\Im (z)<10^{\Re (z)}\}\\
&25^{x+2}-5\cdot9^{x+2}=5^{2x+2}-5\cdot9^{x+1}\\
&3\log_9x^2+\log_{\sqrt{x}}3+7=0\\
&x^4\cdot7^{\log_{\sqrt[3]7}5}\leqslant5^{-\log_{\frac{1}{x}}5}\\
&\begin{cases}
&2^{2x}-3^{2y-1}=-2\\
&2^{2x}+3^{2y}-4\cdot2^x\cdot3^y=-2
\end{cases}\\
&\frac{81^{\frac{1}{\log_5 9}}+27^{\frac{1}{\log_{\sqrt{6}}3}}}{409}\cdot\left( {7^{\frac{1}{\log_{25}7}}-125^{\frac{1}{\log_{\sqrt{6}}5}}}\right)\\
&\frac{1-\ln (x^2+x)}{\ln (x-1)^2 -1} \geqslant -1
\end{align}
\end{document}