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Lecture #19 of Math 110: Logarithmic Functions and Equations, Lecture notes of Elementary Mathematics

Lecture #19 of Math 110, focusing on logarithmic functions, logarithmic equations, and their properties. Students will learn about the Change-of-Base Formula, the product rule, quotient rule, power rule, inverse property, expanding and condensing logarithmic expressions, and solving the simplest logarithmic equations. The lecture includes examples and problems to help students understand these concepts.

Typology: Lecture notes

2021/2022

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Math 110 Lecture #19
CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.
Change-of-Base Formula.
For any logarithmic bases a and b, and any
positive number M,
log
log log
a
b
a
M
Mb
=
Problem #1.
Use your calculator to find the following logarithms.
Show your work with Change-of-Base Formula.
a) b)
2
log 10 1
3
lo
g
9 c)
7
log 11
Using the Change-of-Base Formula, we can graph
Logarithmic Functions with an arbitrary base.
Example:
2
2
ln
log ln 2
log
log log 2
x
x
x
x
=
=
2
log
y
x
=
1
pf3
pf4
pf5

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CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.

ƒ Change-of-Base Formula.

For any logarithmic bases a and b , and any

positive number M ,

log log log

a b a

M

M

b

Problem #1.

Use your calculator to find the following logarithms.

Show your work with Change-of-Base Formula.

a) log 10 2 b) 1

3

log 9 c) log 11 7

ƒ Using the Change-of-Base Formula, we can graph

Logarithmic Functions with an arbitrary base.

Example :

2

2

ln log ln 2

log log log 2

x x

x x

y =log 2 x

CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.

ƒ Properties of Logarithms.

If b, M, and N are positive real numbers, b ≠ 1 , p , x are real

numbers, then

  1. log (^) b MN = log (^) b M + log bN product rule
  2. log (^) b log (^) b log

M

M (^) bN N

= − quotient rule

  1. log log

p b M^ =^ p bM^ power rule

  1. inverse property of logarithms log

log

b , 0

x b

x

b x

b x x

⎪⎩ =^ >

  1. log (^) b M = log bN if and only if M = N.

This property is the base for solving Logarithmic

Equations in form log b g ( x ) = log bh x ( ).

Properties 1-3 may be used for Expanding and Condensing

Logarithmic expressions.

ƒ Expanding and Condensing Logarithmic expressions.

CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.

  1. Solving the Simplest Logarithmic Equation (SLE).

Given: lo g b x = a , b > 0 , b ≠ 1 , a is any real number.

According the definition of the logarithm this equation is

equivalent to

a x = b.

  1. According to properties of logarithms, if

log (^) b M = log bN , then M = N.

Remember, check is part of solution for

Logarithmic Equations.

Problem #4. Solve the following Logarithmic Equations.

a) log 2 x = 5

b) log 3 ( x − 2 )= 5

c) ( )

2 log xx =log 6

d) 1 ( )

2

log x + 4 = − 3

CH. 4.3-4.4 (PART I). Logarithmic Function. Logarithmic equations.

e) log ( x − 15 )= − 2

f) ln ( x + 3 ) = 1

g) log 2( x − 1 ) = log ( x − 2