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Lesson 39: Logarithmic Functions - Natural Logarithms and Properties, Exercises of Algebra

An in-depth explanation of logarithmic functions, focusing on natural logarithms and their properties. It covers the definitions, inverse properties, product property, quotient property, and power property. Students will learn how to expand and condense logarithmic expressions using these properties.

What you will learn

  • What is the definition of a logarithmic function?
  • What is the difference between common and natural logarithms?

Typology: Exercises

2021/2022

Uploaded on 09/12/2022

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MA 22000 Lesson 39 Lesson Notes
(2nd half of text) Section 4.4, Logarithmic Functions
Definition of General Logarithmic Function:
A logarithmic function, denoted by
logb
yx
, is equivalent to
y
bx
.
In previous algebra classes, you may have often used the number 10 as a base for a logarithmic
function. These were called common logarithms. In calculus, the most useful base for
logarithms is the number e. These are called natural logarithms.
Definition of the Natural Logarithmic Function:
The natural logarithmic function is denoted by
lnyx
, which is equivalent to saying
loge
yx
. The function
lnyx
is true if and only if
y
ex
. The natural logarithmic function
is the inverse of the natural exponential function.
The Natural Logarithmic function can be written in logarithmic form or exponential form.
Examine the comparison below.
log
logarithmic form: log exponential form:
ln
logarithmix form: ln exponential form:
y
b
y
b
y
y
y x b x
y x x b
y x e x
y x x e


(For the natural logarithmic function: the x is called the argument, the y is called
the logarithm, and the base is the number e .)
Base Argument Logarithm
pf3
pf4
pf5

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MA 22000 Lesson 39 Lesson Notes (2nd^ half of text) Section 4.4, Logarithmic Functions

Definition of General Logarithmic Function:

A logarithmic function, denoted by y log bx , is equivalent to b yx.

In previous algebra classes, you may have often used the number 10 as a base for a logarithmic function. These were called common logarithms. In calculus, the most useful base for logarithms is the number e. These are called natural logarithms.

Definition of the Natural Logarithmic Function: The natural logarithmic function is denoted by y ln x , which is equivalent to saying y log ex^. The function^ y ln x is true if and only if^ e yx. The natural logarithmic function

is the inverse of the natural exponential function.

The Natural Logarithmic function can be written in logarithmic form or exponential form. Examine the comparison below.

log

logarithmic form: log exponential form:

ln

logarithmix form: ln exponential form:

y b y b y y

y x b x

y x x b

y x e x

y x x e

(For the natural logarithmic function: the x is called the argument, the y is called

the logarithm , and the base is the number e .)

Base Argument Logarithm

Example 1: Convert each logarithmic form to exponential form or each exponential form to logarithmic form.

1 (^4 )

2

7

  1. 3 81 2) 25 5

1

  1. 8 4) ( 4) 64

  2. (2 ) 6) ( 1)

rp

y m

m x

a p e x

   

    

    

2 1 2

5

( 3)

1

  1. log 32 5 8) log 3 8

1

  1. log 5 10) log (2 ) 12 2

  2. log 200 12) ln(3 )

q

x

mn

rs^ x^ u

    (^)     

   

   

Properties of Logarithmic Functions

A Inverse Properties:

ln ln

x x ex ex

(based on inverse property: f ( f ^1 ( )) xx and f ^1 ( f x ( ))  x )

B Product Property: ln^ xy^ ^ ln^ x^ ln y

(based on rule for multiplying powers: bxb y^  bxy )

(^4 2 )

2

  1. ln 20) ln( )

  2. ln 2

xy e x y z

x

x

     

Use the properties A – D to write the expression as the logarithm of a single quantity (condense the logarithmic expression.)

  1. ln ln( 3)

1

  1. ln 3 2ln ln ln( 2) 2

1

  1. ln( 2) 2ln 2ln 4 3

a a

y x x

x x

 

   

  

Solve each equation for x. Give exact answer and approximate answer to 4 decimal places.

22) 3ln 12 23) 2 3 5

x

x

x e

e