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)
  • An additional point

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


E 3.2 Exercises