1. Logarithmic Functions & Their Graphs
3.2

2. Inverses of Exponentials
Remember from section 1.6 that if a function is one-to-one (meaning it passes the Horizontal Line Test) it must have an inverse function. Looking at the graphs of exponential functions of the form f(x) = ax, they all pass the Horizontal Line Test and therefore have an inverse function. The inverse function is called the logarithmic function with base a.

3. Definition of Logarithmic Functions
For x > 0, a > 0, and a ≠ 1 y = loga x If and only if x = ay. The function given by f(x) = loga x is called the logarithmic function with base a. Read as log base a of x.

4. When evaluating logarithms, remember that loga x is the exponent to which a must be raised to obtain x. For example: log2 8 = 3 because 23 = 8.

5. Example 1: Evaluating Logarithms
Use the definition of logarithmic functions to evaluate each logarithm at the indicated value of x.
f(x) = log2 x x = 32
F(x) = log3 x x = 1
F(x) = log4 x x = 2
F(x) = log10 x x =

6. Common Logarithm
The logarithmic function with base 10 is called the common logarithmic function.

7. Example 2: Evaluating Common Logarithms on a Calculator
Use a calculator to evaluate the function f(x) = log10 x at each value of x.
x = 10
x = 2.5
x = -2
x =

8. Error Message
In part c we obtain an error message because the domain of every logarithmic function is the set of positive real numbers.

9. Properties of Logarithms
loga 1 = 0 because a0 = 1
loga a = 1 because a1 = a
loga ax = x and alogx = x INVERSES
If loga x = loga y then x = y

10. Example 3: Using Properties of Logarithms
Solve for x: log2 x = log2 3
Solve for x: log4 4 = x
Simplify: log5 5x
Simplify: 7log7 14

11. The Natural Logarithmic Function
The function f(x) = ex is one-to-one and has an inverse function. This inverse function is called the natural logarithmic function and is denoted by ln x (read as the natural log of x).

12. For x > 0, y = ln x if and only if x = ey
For x > 0, y = ln x if and only if x = ey. The function given by f(x) = loge x = ln x is called the natural logarithmic function. NOTE: the natural logarithm ln x is written without a base. The base is understood to be e.

13. Example 4: Evaluating the Natural Logarithmic Function
Use a calculator to evaluate the function f(x) = ln x at each indicated value of x.
x = 2
x = .3
x = -1

14. Properties of Natural Logarithms
ln 1 = 0 because e0 = 1
ln e = 1 because e1 = e
ln ex = x and elnx = x INVERSES
If ln x = ln y, then x = y

15. Example 5: Using Properties of Natural Logarithms
Use the properties of natural logarithms to rewrite each expression. ln
eln 5
ln e0
2 ln e

16. Example 6: Finding the Domains of Logarithmic Functions
Find the domain of each function.
f(x) = ln (x – 2)
g(x) = ln (2 – x)
h(x) = ln x2