Properties of logarithmic function and its graph

The function $y=\log_a x$

Domain: $x>0$ . Range: all real $y$ . It increases if $a>1$ , decreases if $0<a<1$ ; the point $(1;0)$ always lies on the graph. Vertical asymptote $x=0$ (the branch goes to $-\infty$ or $+\infty$ depending on $a$ as $x\to0^+$ ). Together with $y=a^x$ it is the inverse function: symmetry about $y=x$ .

In problems with a parameter, monotonicity of $\log_a$ sets the direction of inequalities

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