Sketching logarithmic graphs

The curve \(y=A\log_a(x-h)+c\) exists only where its argument is positive. Its asymptote is vertical, at the value of \(x\) that makes the argument zero.

Method

1. Asymptote: put the argument equal to zero, draw the dashed line \(x=h\) and label it.
2. Domain: the argument must be positive, so the curve lies on one side of the asymptote only.
3. \(x\)-intercept: put \(y=0\) and change to index form.
4. \(y\)-intercept: put \(x=0\), only if \(0\) is in the domain.
5. Shape: for a base greater than 1, the curve rises when \(A>0\) and falls when \(A<0\). Plot one further point, then draw the curve close to the asymptote.

Example

Sketch \(y=-2\lg(x+5)\).

\(\begin{aligned} x+5=0&\quad\Rightarrow\quad\text{asymptote}\hspace{0.5em}x=-5,\ \text{domain}\hspace{0.5em}x>-5 \\ y=0:\quad\lg(x+5)=0&\quad\Rightarrow\quad x+5=1,\ x=-4 \\ x=0:\quad y&=-2\lg5\approx-1.40 \\ x=5:\quad y&=-2\lg10=-2 \end{aligned}\)

Since \(A=-2<0\), the curve falls from left to right, and it rises steeply close to the asymptote.