Logarithmic Functions

Section 2

Objectives

• Change Exponential Expressions to Logarithmic Expressions and Logarithmic Expressions to Exponential Expressions

• Evaluate Logarithmic Expressions • Determine the Domain of a Logarithmic

Function• Graph Logarithmic Functions• Solve Logarithmic Equations

where a > 0 and a ≠ 1

Domain: x > 0€

y = loga x if and only if x = ay

Change to an equivalent logarithmic function

€

1.23 = m

eb = 9

a4 = 24

Change to an equivalent exponential function

€

loga 4 = 5

loge b = −3

log3 5 = c

Find the exact value of the logarithmic expression

€

log216

log3

1

27

Since logarithmic and exponential functions are inverses of each other,

Domain of the logarithmic fx. = Range of the exponential fx. = (0, ∞)

Range of the logarithmic fx. = Domain of the exponential fx. = (-∞, ∞)

The argument of a logarithmic function must be greater than zero

Find the domain of the logarithmic function

€

f (x) = log2(x + 3)

g(x) = log5

1+ x

1− x

⎛

⎝ ⎜

⎞

⎠ ⎟

h(x) = log1

2

x

Properties of the Logarithmic Function f(x) = logax

• Domain: Positive reals Range: All reals• x-intercept: 1 y-intercept: None• y-axis is a vertical asymptote• Decreasing if 0 < a < 1 and increasing if a > 1• Graph contains the points (1, 0), (a, 1), and

(1/a, -1)• Graph is smooth and continuous, with no corners

or gaps

y = ln x if and only if x = ey

(inverse functions)

========================

y = log x if and only if x = 10y

(inverse functions)

Solve the logarithmic equation.

€

log3(4x − 7) = 2

Solve the logarithmic equation.

€

logx 64 = 2

Use a logarithm to solve the equation.

€

e2x = 5

Pages 515-516 (22-56 even)

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Pages 517-518 (76-110 even, 118, 121)