Logarithmic function y = logₐx. Graphs and properties

Logarithmic function $y=\log_a x$

By a logarithmic function we mean $f(x)=\log_a x$ with $a>0$ , $a\neq1$ — a fixed base. Domain: $D(f)=(0;+\infty)$ — only positive arguments are allowed under the log. Range: $E(f)=\mathbb{R}$ — every real value occurs as the logarithm of some $x>0$ . Anchor point: $(1;0)$ , since $\log_a 1=0$ . Vertical asymptote: the line $x=0$ (the branch goes up or down as $x\to 0^+$ depending on $a$ ). Monotonicity: if $a>1$ the function strictly increases on $(0;+\infty)$ ; if $0<a<1$ it strictly decreases.

The graph always lies to the right of the $Oy$ axis — only for $x>0$

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