Elective Math Topics for Year 9

1. ©Carolyn C. Wheater, 2000 1
Exponents and Logarithms
 Definition of a Logarithm
 Rules
 Functions
 Graphs
 Solving Equations

2. ©Carolyn C. Wheater, 2000 2
Definition of a Logarithm
A logarithm, or log, is defined
in terms of an exponential.
If bx=a, then logba=x
 If 52=25 then log525=2
 log525=2 is read “the log base 5
of 25 is 2.”
 You might say the log is the
exponent we put on 5 to make 25

3. ©Carolyn C. Wheater, 2000 3
Rules for Exponents
 Exponents give us many shortcuts
for multiplying and dividing quickly.
 Each of the key rules for exponents
has an important parallel in the world
of logarithms.

4. ©Carolyn C. Wheater, 2000 4
Multiplying with
Exponents
To multiply powers of the same base,
keep the base and add the exponents.
x y x x x y
x y x y
7 3 5 7 5 3
7 5 3 12 3

  

Keep x, add
exponents 7 + 5
Can’t do anything
about the y3 because
it’s not the same base.

5. ©Carolyn C. Wheater, 2000 5
Dividing with Exponents
7 5
7 5
7
7
5
5
7 5 7 5
10 12
6 4
10
6
12
4
10 6 12 4 4 8


 
   
 
To divide powers of the same base, keep
the base and subtract the exponents.
Keep 7,
subtract 10-6
Keep 5,
subtract 12-4

6. ©Carolyn C. Wheater, 2000 6
Powers with Exponents
To raise a power to a power, keep the base
and multiply the exponents.
This means t7·t7·t7
= t7+7+7

7. ©Carolyn C. Wheater, 2000 7
Rules for Logarithms
 Just as the rules for exponents let you
easily rewrite a product, quotient, or
power, the corresponding rules for logs
allow you to rewrite the log of a
product, the log of a quotient, or the
log of a power.

8. ©Carolyn C. Wheater, 2000 8
Log of a Product
Logs are exponents in disguise
 To multiply powers, add exponents
 To find the log of a product, add the logs of the
factors
The log of a product is the sum of the logs
of the factors
logbxy = logbx + logby
log5(25·125) = log525 + log5125

9. ©Carolyn C. Wheater, 2000 9
Log of a Product
Think about it:
 25·125 = 52 ·53 = 52+3=55
 log5(25 ·125) = log5(52 ·53)=log5(52)+log5(53)
 log525 = log5(52)=2
 log5125 = log5(53)=3
 log5(25 ·125) = log5(52)+log5(53) = 2 + 3 = 5
 log5(25 ·125) = log5(55) =5
Laws of Exponents
Logs are Exponents!
Add the exponents!

10. ©Carolyn C. Wheater, 2000 10
Log of a Quotient
Logs are exponents
 To divide powers, subtract exponents
 To find the log of a quotient, subtract the
logs
The log of a quotient is the difference
of the logs of the factors
 logb = logbx - logby
 log5(12525) = log5125 - log525
x
y

11. ©Carolyn C. Wheater, 2000 11
Log of a Quotient
Think about it:
 125  25 = 53  52 = 53-2=51
 log5(125  25) = log5(53  52) = log5(53) - log5(52)
 log5125 = log5(53)=3
 log525 = log5(52)=2
 log5(125  125) = log5(53)-log5(52) = 3 - 2 = 1
 log5(125  25) = log5(51) =1
Laws of Exponents
Logs are Exponents!
Subtract the
exponents!

12. ©Carolyn C. Wheater, 2000 12
Log of a Power
Logs are exponents
 To raise a power to a power, multiply
exponents
 To find the log of a power, multiply the
exponent by the log of the base
The log of a power is the product of the
exponent and the log of the base
 logbxn = nlogbx
 log 32 = 2log3

13. ©Carolyn C. Wheater, 2000 13
Log of a Power
Think about it:
 252 =( 52)2 = 52 · 2=54
 log5(252) = 2log5(52)
 log525 = log5(52)=2
log5(252) = 2log5(52) = 2 ·2 = 4
 log5(252) = log5(625) = log5(54) = 4
Laws of Exponents
Logs are Exponents!
Multiply the exponent
by the log (an
exponent!)

14. ©Carolyn C. Wheater, 2000 14
Rules for Logarithms
 The same rules can be used to turn an
expression into a single log.
 logbx + logby = logbxy
 logbx - logby = logb
 nlogbx = logbxn
x
y

15. ©Carolyn C. Wheater, 2000 15
Rules for Logarithms
 A sum of two logs becomes the log of a
product.
 log39 + log327 = log3(9·27)
 A difference of two logs becomes the log of
a quotient.
 log232 - log28 = log2
 A multiple of a log becomes the log of a
power
 2log57 = log572
32
8
Bases must be
the same

16. ©Carolyn C. Wheater, 2000 16
Sample Problem
 Express as a single logarithm:
3log7x + log7(x+1) - 2log7(x+2)
 3log7x = log7x3
 2log7(x+2) = log7(x+2)2
log7x3 + log7(x+1) - log7(x+2)2
 log7x3 + log7(x+1) = log7(x3·(x+1))
log7(x3·(x+1)) - log7(x+2)2
 log7(x3·(x+1)) - log7(x+2)2 =

17. ©Carolyn C. Wheater, 2000 17
Exponential Functions
The exponential function has the form
f(x)=abx
 a is the beginning, or initial amount
 b is the base, the factor that represents the
rate of increase
 x is the exponent, often representing a
period of time

18. ©Carolyn C. Wheater, 2000 18
Logarithmic Functions
The logarithmic function has the form
f(x)=logbx
 b is the base
 x is the number
 f(x) is the log (or disguised exponent)

19. ©Carolyn C. Wheater, 2000 19
Graphs of Exponential
Functions
The graph of f(x)=bx
has a characteristic shape.
 If b>1, the graph rises
quickly.
 If 0 < b < 1, the graph falls
quickly.
 Unless translated the graph
has a y-intercept of 1.
24

20. ©Carolyn C. Wheater, 2000 20
Graphs of Logarithmic
Functions
The graph of f(x)=logbx
has a characteristic
shape.
 The domain of the
function is {x| x>0}
 Unless translated, the
graph has an x-intercept
of 1.
1 1 2 3 4 5 6

21. ©Carolyn C. Wheater, 2000 21
Translating the Graphs
Both exponential and logarithmic functions
can be translated.
The vertical and horizontal slides will show
up in predictable places in the equation, just
as for parabolas and other functions.
f x x h k
f x x
b
( ) log ( )
( ) log ( )
  
  
3 6 4
Shifted 1 unit right
and 3 down
Shifted 6 units left
and 4 up

22. ©Carolyn C. Wheater, 2000 22
Solving Exponential
Equations
 If possible, express
both sides as powers
of the same base
Equate the exponents
Solve
x x
  
4 4 14
27 3 9
3 3 3
3 3
3 3
1 2 7
3 1 2 2 7
3 1 2 2 7
4 4 14
( )
( ) ( )
( )
x x
x x
x x
x x
 
 
  
 




4 3 14
18 3
6
 


x
x
x

23. ©Carolyn C. Wheater, 2000 23
Solving Exponential
Equations
 If it is not possible to express both sides
as powers of the same base
 take the log of each side using any
convenient base
 use rules for logs to break down the
expressions
 isolate the variable
 evaluate and check

24. ©Carolyn C. Wheater, 2000 24
Solving Exponential
Equations
 Solve
 Take the log of each side
 Use rules for logs
Isolate the variable
 Evaluate and check
5 7
5 7
3 5 1 7
3 5 7 7
3 5 7 7
3 5 7 7
7
3 5 7
3 1
3 1
x x
b
x
b
x
b b
b b b
b b b
b b b
b
b b
x x
x x
x x
x
x


 
 
 
 




log ( ) log ( )
log ( )log
log log log
log log log
( log log ) log
log
log log
Any convenient base can be used,
and since you’ll want to use your
calculator, that will probably be 10
x  0.675

25. ©Carolyn C. Wheater, 2000 25
Solving Logarithmic
Equations
Use the rules for logs to simplify each side
of the equation until it is a single log or a
constant.
log log log ( )
log log log ( )
log ( ) log ( )
log ( ) log ( )
2 2 2
2 2
2
2
2
2
2
2
2
2 2
2
2 5 2 6
5 6
5 6
25 6
x x
x x
x x
x x
  
  
  
 
log log log
log log log ( )
log log
log
5 5 5
5 5
2
5
3
5 5
5
2 7 125
7 5
49 3
49
3
x
x
x
x
 
 
 


26. ©Carolyn C. Wheater, 2000 26
Solving Logarithmic
Equations
Log = Log
Exponentiate (drop logs)
 Solve the resulting
equation
 Reject solutions that
would mean taking the
log of a negative number
log log log ( )
log ( ) log ( )
( )
( )( )
2 2 2
2 2
2
2
2
2
2 5 2 6
25 6
25 6
25 12 36
0 37 36
0 36 1
36 0 1 0
36 1
x x
x x
x x
x x x
x x
x x
x x
x x
  
 
 
  
  
  
   
 

27. ©Carolyn C. Wheater, 2000 27
Solving Logarithmic
Equations
 Log = Constant
Use the definition of
a logarithm to
express as an
exponential
Evaluate and check
log log log
log
5 5 5
5
3
2 7 125
49
3
5
49
125
49
125 49 6125
x
x
x
x
x
 



  