1. 4.4 Logarithmic Functions
The function f (x) = ax, a  1, is one-to-one and thus has an inverse.
The logarithmic function with base a and the exponential function with base a are inverse functions. So,

2. 4.4 Graphs of Logarithmic Functions
Recall that the graph of the inverse function is reflexive about the line y = x.
The figure above is the typical shape for such graphs where a > 1 (includes base e and base 10 graphs).

3. 4.4 Graphs of Logarithmic Functions
Below are typical shapes for such graphs where
0 < a < 1.

4. 4.4 The Logarithmic Function: f (x) = loga x, a > 1

5. 4.4 The Logarithmic Function: f (x) = loga x, 0 < a < 1

6. 4.4 Determining Domains of Logarithmic Functions
Example Find the domain of each function.
Solution
Argument of the logarithm must be positive.
x – 1 > 0, or x > 1. The domain is (1,).
Use the sign graph to solve x2 – 4 > 0.
The domain is
(–,–2) (2, ).

7. 4.4 Graphing Translated Logarithmic Functions
Example Give the domain, range, asymptote, and
x-intercept.
(a)
Solution
The argument x – 1 shifts the graph of y = log2 x 1 unit to the right.
Vertical asymptote: x = 1
x-intercept: (2,0)
Domain: (1,), Range: (–, )

8. 4.4 Graphing Translated Logarithmic Functions
Here, 1 is subtracted from y = log3 x shifting it down 1 unit.
Vertical asymptote: y-axis (or x = 0)
x-intercept : (3,0)
Domain: (0,), Range: (–, )

9. 4.4 Determining Symmetry Example Show analytically that the graph of
is symmetric with respect to the
y-axis.
Solution
Since f (x) = f (–x), the graph is symmetric with
respect to the y-axis.

10. 4.4 Finding the Inverse of an Exponential Function
Example Find the inverse function of
Solution
Replace f (x) with y.
Interchange x and y.
Isolate the exponential.
Write in logarithmic form.
Replace y with f –1(x).

11. 4.4 Logarithmic Model: Modeling Drug Concentration
Example The concentration of a drug injected into the
bloodstream decreases with time. The intervals of time in
hours when the drug should be administered are given by
where k is a constant determined by the drug in use, C2 is the
concentration at which the drug is harmful, and C1 is the
concentration below which the drug is ineffective. Thus, if
T = 4, the drug should be administered every 4 hours. For a
certain drug, k = C2 = 5, and C1 = 2. How often should the
drug be administered?

12. 4.4 Logarithmic Model: Modeling Drug Concentration
Solution Substitute the given values into the
equation.
The drug should be given about every 3 hours.