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

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

5. Dividing with Exponents
To divide powers of the same base, keep the base and subtract the exponents.
Keep 5, subtract 12-4
Keep 7, subtract 10-6
©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
©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.
©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
©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) = = 5
log5(25 ·125) = log5(55) =5
Laws of Exponents
Logs are Exponents!
Add the exponents!
©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) = log log525
©Carolyn C. Wheater, 2000

11. Subtract the exponents!
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) = = 1
log5(125  25) = log5(51) =1
Laws of Exponents
Logs are Exponents!
Subtract the exponents!
©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
©Carolyn C. Wheater, 2000

13. Multiply the exponent by the log (an exponent!)
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!)
©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
©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
Bases must be the same
©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 =
©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
©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)
©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
©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
©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.
Shifted 1 unit right and 3 down
Shifted 6 units left and 4 up
©Carolyn C. Wheater, 2000

22. Solving Exponential Equations
If possible, express both sides as powers of the same base
Equate the exponents
Solve
©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
©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
Any convenient base can be used, and since you’ll want to use your calculator, that will probably be 10
x  0.675
©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.
©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
©Carolyn C. Wheater, 2000

27. Solving Logarithmic Equations
Log = Constant
Use the definition of a logarithm to express as an exponential
Evaluate and check
©Carolyn C. Wheater, 2000