# How do you graph g(x) = log_3(x - 4)?

Jun 3, 2016

Make a smooth graph that is gently-sloping-down to horizontal through (35/9, -2), (11/3, -1), (5, 0), (7, 1), (13, 2), ( (31, 3),...,(4+3^N, N)... x = 4 ( downwards ) is its vertical asymptote,

#### Explanation:

The inverse relation for $y = g \left(x\right) = {\log}_{3} \left(x - 4\right)$ is

$x - 4 = {3}^{y}$.

So, x = 4 + 3^y#.

As $y \to - \infty , x \to 4$. So, x =4 is the vertical asymptote to the graph.

As $x \to \infty , y \to \infty , \mathmr{and} \frac{d}{\mathrm{dx}} \left(y\right) = \frac{1}{x - 4} \to 0$.

A short Table for making graph-by-hand, on a rectangular graph paper, is

$\left(x . y\right) : \left(\frac{35}{9} , - 2\right) , \left(\frac{11}{3} , - 1\right) , \left(5 , 0\right) , \left(7 , 1\right) , \left(13 , 2\right) , \left(31 , 3\right) , \ldots , \left(4 + {3}^{N} , N\right) , \ldots$