HPC-A Unit-1 Lesson 3 NOTES:

A – (3-2): Logarithm Functions

3 – 2 Objectives

Definition of logarithm functions

Changing exponential functions to logarithm functions.

Evaluation of logarithm functionswithout the use of a calculator

Properties of logarithm functions

Graphing logarithms functions

I) A logarithm function is a function in form of

, (log of x to base b) wherex> 0; b > 0 and b ≠ 1

II) It is always easy to change from an exponential function to logarithm function because

is equivalent to

NOTE: i) Given a log. function without a base, means it’s base is 10 (i.e )

ii) and (Basic logarithm property)

iii) and (Inverse property oflogarithm)

Examples:

i) Change the following logarithm functions to exponential functions

a) b)

ii) Change the following logarithm functions to exponential functions and solve for the

variables

a) b)

iii) Use properties of logarithm functions to evaluate the following:

a) b)

III) Characteristics of exponential function (see pg. 379):

a)It’s domain consists of set of positivenumbers:; and range are set real

numbers:.

b) It’s graph passes through point (1, 0)the x-intercept and there is no y-intercept.

c) The function is increasing if, and decreasing if,

NOTE: i) Given, means domain isand vertical asymptote is x=h

ii) The logarithm function with a base e is called natural logarithm function

Properties of natural logarithm

General properties Natural logarithmproperties

a)

b)

c)

d)

IV) Transformations of exponential functions: The parent function is (see pg. 380):

Transformation / Graph / Effects on graph
Vertical Shifts /
/ Graph shifts up c units
Graph shits down c units
Horizontal Shifts /
/ Graph shifts to the left c units
Graph shifts to the right c units
Vertical asymptote: x = c
Reflection /
/ Graph reflected about the x-axis (i.e multiply y by )
Graph reflected about the y-axis (i.e divide x by )
Vertical stretching or
Shrinking / / Graph stretches vertically if c > 1 and
shrinks vertically if 0 < c < 1. (i.e multiply y by c)
Horizontal stretching
orShrinking / / Graph stretches horizontally if 0 < c < 1. and
shrinks horizontally if c > 1. (i.e dividex by c)

NOTE: When graphing logarithm functions always switch x and y first.

Examples:

Graph the following functions. For each graph, you must show at least 5 points.

  1. 2)

3) 4)

B – 3-3 Properties of Logarithms

Properties of logarithms

Let b, M, and N be positive real numbers with

i) -- Product Rule

ii) -- Quotient Rule

iii) -- Power Rule

NOTE: These properties apply to both expanding and condensing logarithms exponents.

Examples

A) Expand and simplify

1) 2) 3)

4) 5) 6)

B) Condense the expressions below:

1) 2) 3)

C) Change of base property:

For any logarithmic bases a and b, and any positive number M,

, when using the calculator and

example:

1)Evaluate

C - (3-4): Exponential & LogarithmsEquations

The following techniques can be used to solve exponential functions

I) Expressing both sides in the same base:

a) Rewrite as bm= bn

b) Set m = n

c) Solve for the variable

II) Using natural log by isolating the exponential expression:

a) Take natural of both sides and simplify (NOTE: or )

b) Solve for the variable

III) By using factoring techniques

Examples:

1) 2)

3) 4)

5) 6)

7) 8)

9) 10)

When solving for logarithmic equations we need to:

a) Express the equation in the form of

b) Use the definition of logarithm to rewrite the equation in exponential form

i.e means

c) Solve for the variable and check for the values for which M > 0

NOTE: Be sure you know how to determine the domain of logarithmic functions

Examples:

1) 2) 3)

4) 5)