FINITE MATH
1.6 Notes – LOGARITHMIC FUNCTIONS
The word “logarithm” (abbreviated “log”) means power or exponent. The number being raised to the power is called the base and is written as a subscript. log 1000
10
means “the exponent to which we have to raise 10 to get 1000. Since
10
3 
1000
, the exponent is 3, so the logarithm is 3. Therefore, log 1000
10
= 3
COMMON LOGARITHM
Logarithm with base 10
log x
 y
is equivalent to
10 y  x
(since log x
 y
really means log
10 x
 y
Is defined only for positive values of x.
Examples:
(a) Evaluate log 100
)
(b) Evaluate log 1
10
(c) Find the two logarithms above on your calculator.
PROPERTIES OF COMMON LOGARITHMS
1.
log1

0
5. log( M N )
 log M
 log N
2.
log10

1
6. log

1
N

  
  log N
3.
log10 x  x
7. log

M
N

 
  log M
 log N
4.
10 log x  x
8. log( M
P
) log M
**LOGARITHMS BRING DOWN EXPONENTS!**
EXAMPLES: Use the properties of logarithms to simplify and tell which property you
used. a.
log10
7  _____ Property:____________ b.
log log13  _____ Property:____________ c.
log(3 4)
 ___________________________ Property:_____________ d.
e.
f.
log
1
 
___________________________ Property:_____________ log
3
 
3 log(5 )

___________________________ Property:_____________
___________________________ Property:_____________

Example: Finding When a Car Depreciates by Half
A car depreciates by 10% per year. When will it be worth only half its original
value?
GRAPHS OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS
If the point (x, y) lies on the graph of y = log x, then x

10 y
An inverse is found by reversing x and y. Therefore, y

10 x
. is the inverse of x

10 y
, and the point (y, x) will lie on the graph of
the curves y = x and y

10 x y

10 x . Therefore,
are related by having their
x- and y-coordinates reversed, so the curves are mirror images of
each other across the line y = x. (See graph below.)
Since the point (0, 1) lies on the graph of the exponential function of the form
( )
 b x
, the point (1, 0) lies on the graph of
( )
 log b x
.
-
( )
 b x and
( )
 log b x
are inverse functions.
LOGARITHMS TO OTHER BASES
Any positive number other than 1 may be used as a base for logs, using the following definition:
Base a Logarithms: log a x
 y
is equivalent to a y  x
.

NATURAL LOGARITHMS
Logarithms to the base e are called natural logarithms.
The natural logarithms of a positive number x is written ln x
and may be found using the LN key on a calculator.
DEFINITION: Natural Logarithms ln x
 y
is equivalent to e y  x
.
Examples:
(a) log 27
3
 ______
(b) log 5
25
 ______
(c) Use a calculator to find: ln 9.57 = ____________
PROPERTIES OF NATURAL LOGARITHMS
1.
ln 1 = 0 5. ln( M N )
 ln M
 ln N
2.
3.
ln ln e
4.
e ln x e x
= 1
 x
 x
6.
7.
8. ln ln ln(

1
N

  
  ln N

M
N

 
 
M
P
) ln M

  ln M ln N
Examples: Using the Properties of Natural Logarithms
(a) ln e
7  _____ (d) ln
1
 
______________ = ________
(b)
(c) e ln13  ln(3 4)
_____

(e) ln
3
 
___________________
________________ (f)
3 ln(5 )
 ____________
Examples: Simplify the following logarithmic functions.
(a)
( )
 x

(b)
( )
 ln x
 

1
(c) f x
 x
5  x
3
( ) ln( ) ln( )

APPLICATION: Dating by Carbon-14
All living things absorb small amounts of radioactive carbon-14 from the atmosphere.
When they die, the carbon-14 stops being absorbed and decays exponentially into ordinary carbon. Therefore, the proportion of carbon-14 still present in a fossil or other ancient remain can be used to estimate how old it is. The proportion of the original carbon-14 that will be present after t years is
(Proportion of carbon-14 remaining after t years) = e

0.00012
t
The Dead Sea Scrolls, discovered in a cave near the Dead Sea in what was then Jordan, are among the earliest documents of Western civilization. Estimate the age of the Dead
Sea Scrolls if the animal skins on which some were written contains 78% of their original carbon-14.
,