# 3.2 | Logarithmic Functions and Their Graphs

For $$x >0, a >0,$$ and $$a \not = 1,$$
$$y = \log_a x \text{ if and only if } x = a^y.$$
The function given by
$$f(x) = \log_a x$$
is called the logarithmic function with base $$a.$$
The function defined by
$$f(x) = \log_e x = \ln x,$$
where $$x >0$$ is called a natural logarithmic function.
In general, to graph
$$f(x) = a \log_b(kx-c) + d$$
you can rely on three defining characteristics of the function:

• The vertical asymptote: $$bx-c = 0$$
• The $$x$$-intercept (one always exists)

Plot the vertical asymptote using a dashed line, plot the $$x$$-intercept and an additional point. Use a smooth curve to connect the two points and extend the graph so that the curve becomes asymptotic to the line $$x = c/b$$. You should also extend the curve in the other direction and so that the curve moves away from the line $$x = c/b$$. Essentially you want to preserve the general shape of the basic logarithmic function, when graphing its variations.

Properties of Logarithms

1. $$\log_a 1 = 0$$
2. $$\log_a a = 1$$
3. Inverse Property: $$\log_a a^x = x$$ and $$a^{\log_a x} = x$$
4. One-to-One Property: If $$\log_a x = \log_a y,$$ then $$x = y.$$
Properties of Logarithms

1. $$\ln 1 = 0$$
2. $$\ln e = 1$$
3. Inverse Property: $$\ln e^x = x$$ and $$e^{\ln x} = x$$
4. One-to-One Property: If $$\ln x = \ln y,$$ then $$x = y.$$

Graph
$$f(x) = 2\log_2(x-1) -1$$