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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.
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Typology: Exercises
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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 y x.
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 y x. 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.
y b y b y y
Base Argument Logarithm
Example 1: Convert each logarithmic form to exponential form or each exponential form to logarithmic form.
1 (^4 )
2
7
1
8 4) ( 4) 64
(2 ) 6) ( 1)
rp
y m
m x
a p e x
2 1 2
5
( 3)
1
1
log 5 10) log (2 ) 12 2
log 200 12) ln(3 )
q
x
mn
rs^ x^ u
(^)
Properties of Logarithmic Functions
A Inverse Properties:
ln ln
x x e x e x
(based on inverse property: f ( f ^1 ( )) x x and f ^1 ( f x ( )) x )
B Product Property: ln^ xy^ ^ ln^ x^ ln y
(based on rule for multiplying powers: bx b y^ bx y )
(^4 2 )
2
ln 20) ln( )
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
1
a a
y x x
x x
Solve each equation for x. Give exact answer and approximate answer to 4 decimal places.