3.2

Logarithmic Functions & Their Graphs

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.

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.

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.

Example 1: Evaluating Logarithms

Use the definition of logarithmic functions to evaluate each logarithm at the indicated value of x.

A)f(x) = log2 x x = 32

B)F(x) = log3 x x = 1

C)F(x) = log4 x x = 2

D)F(x) = log10 x x = 1001

Common Logarithm

The logarithmic function with base 10 is called the common logarithmic function.

Example 2: Evaluating Common Logarithms on a Calculator

Use a calculator to evaluate the function f(x) = log10 x at each value of x.

A)x = 10

B)x = 2.5

C)x = -2

D)x = 41

Error Message

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

Properties of Logarithms

1. loga 1 = 0 because a0 = 1

2. loga a = 1 because a1 = a

3. loga ax = x and alogx = x INVERSES

4. If loga x = loga y then x = y

Example 3: Using Properties of Logarithms

A) Solve for x: log2 x = log2 3

B) Solve for x: log4 4 = x

C) Simplify: log5 5x

D) Simplify: 7log7 14

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).

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.

Example 4: Evaluating the Natural Logarithmic Function

Use a calculator to evaluate the function f(x) = ln x at each indicated value of x.

A)x = 2

B)x = .3

C)x = -1

Properties of Natural Logarithms

1. ln 1 = 0 because e0 = 1

2. ln e = 1 because e1 = e

3. ln ex = x and elnx = x INVERSES

4. If ln x = ln y, then x = y

Example 5: Using Properties of Natural Logarithms

Use the properties of natural logarithms to rewrite each expression.

A)ln

B)eln 5

C)ln e0

D)2 ln e

e

1

Example 6: Finding the Domains of Logarithmic Functions

Find the domain of each function.

A)f(x) = ln (x – 2)

B)g(x) = ln (2 – x)

C)h(x) = ln x2