Section 5.4 logarithmic functions

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Section 5.4
Logarithmic Functions

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( ) ( )3 2
1
a log 81 b log
8
(a) 3 raised to what power yields 81? 3log 81
3 81y
y =
=
4
3 3
4
y
y
=
= 3Therefore, log 81 4=
3
2 2
3
y
y
−
=
= − 2
1
Therefore, log 3
8
= −
(b) 2 raised to what power yields ?
1
8
2
3
3
1
log
8
1 1
2 2
8 2
y
y
−
=
= = =

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( ) ( ) ( ) ( ) ( )
( ) ( )
3 2
2
3
a log 2 b log
1
c log 1
x
f x x F x
x
h x x
+ 
= − =  ÷
− 
= −
( )a The domain of consists of all for which 2 0.f x x − >
( )2 or 2,x > ∞
( )
3
b The domain of is restricted to 0
1
x
F
x
+ 
> ÷
− 
( ) ( ), 3 1, .−∞ − ∪ ∞
( )c Since the absolute value function is never negative, the domain
would consist of all real numbers except 1 0.x − =
( ) ( ),1 1,−∞ ∪ ∞

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( ) ( ) ( )2Solve: log 2 1 3 log 343 3xa x b+ = =
( )2(a) Change log 2 1 3 to exponential form.x + =
3
2 2 1x= + 8 2 1x= +
7
2
x =
2 2
7
log 2 1 log 8 3
2
  
+ = = ÷ ÷
  
(b) Change log 343 3 to exponential form.x =
3
343x = 7x =
7log 343 3=

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3
Solve: 2 6x
e =
3
3x
e = Isolate the exponential.
ln3 3x= Change to logarithmic form.
ln3
3
x = Exact solution
0.366≈ Approximate solution