Graphing Logarithmic Functions — Cheat sheet

The shape

$y = \log_a x$: domain $x > 0$, range every real $y$, vertical asymptote $x = 0$.

The two are mirror images in the $x$-axis, because $\log_{1/a} x = -\log_a x$.

Three points fix the curve

For $y = \log_2 x$: $\left(\dfrac{1}{2}, -1\right)$, $(1, 0)$, $(2, 1)$. Plot them, follow the shape, done.

$(1, 0)$ is on every logarithm graph, whatever the base.

Reflection in y = x

Every row is the row above with the coordinates swapped.

Shifts

• asymptote $x = h$, domain $x > h$ — the inside constant moves the wall
• $+k$ lifts the curve and leaves the wall alone

Method for a shifted graph

Take $f(x) = \log_2(x - 3) + 1$:

1. Wall where the argument is zero: $x = 3$, so domain $x > 3$
2. Argument $= 1$ at $x = 4$: $f(4) = 0 + 1 = 1$
3. Argument $= 2$ at $x = 5$: $f(5) = 1 + 1 = 2$
4. Base above $1$, so it rises from the wall through $(4, 1)$ and $(5, 2)$

Traps

• A negative height never means a negative $x$. $\log_3 x = -2$ gives $x = \dfrac{1}{9}$, not $-9$.
• Heights are negative exactly on $0 < x < 1$.
• The curve is unbounded below near the wall, but never crosses it.
• $\log_2 x$ gains one unit of height each time $x$ doubles.