Logarithmic Expressions and Functions

The definition is a translation rule

$\log_b c = a$ means exactly $b^a = c$. Every logarithm question can be attacked by translating to exponential form and back. Because $b^a>0$ for any real $a$, the input of a logarithm must be strictly positive — this single fact generates the domain restrictions and the extraneous solutions of the whole unit.

The function $y=\log_b x$

As the inverse of $y=b^x$ (with $b>0$, $b\neq1$), the log function inherits everything by reflection across $y=x$:

• domain $(0,\infty)$, range $(-\infty,\infty)$;
• a vertical asymptote at $x=0$ — the mirror image of the exponential's horizontal asymptote $y=0$;
• key points $(1,0)$ and $(b,1)$, the reflections of $(0,1)$ and $(1,b)$;
• increasing without bound, but slower than any polynomial: the graph keeps rising forever, just agonizingly slowly. There is no horizontal asymptote.

Transformations move the asymptote with the graph: $y=\log_b(x-h)+k$ has vertical asymptote $x=h$. When the inside is $g(x)$ rather than $x$, solve $g(x)=0$ for the asymptote and $g(x)>0$ for the domain.