The document discusses properties of logarithms. It begins by recalling rules of exponents and their corresponding rules of logarithms. Four basic logarithm rules are presented: 1) logb(1) = 0, 2) logb(xy) = logb(x) + logb(y), 3) logb(x/y) = logb(x) - logb(y), 4) logb(xt) = tlogb(x). It then works through an example problem to demonstrate using these rules to write the logarithm of a expression in terms of logarithms of its variables. It concludes by noting that logarithms and exponentials are inverse functions, so logb(bx) =

1. Properties of Logarithm

2. Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

3. 1. b0
= 1
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

4. 1. b0
= 1
2. br
· bt
= br+t
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

5. 1. b0
= 1
2. br
· bt
= br+t
3. = br-t
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

6. 1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

7. 1. logb(1) = 01. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

8. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

9. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

10. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:

11. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:

12. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers.

13. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let logb(x) = r
and logb(y) = t, which in exp-form are x = br
and y = bt
.

14. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let logb(x) = r
and logb(y) = t, which in exp-form are x = br
and y = bt
.
Therefore x·y = br+t
,

15. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let logb(x) = r
and logb(y) = t, which in exp-form are x = br
and y = bt
.
Therefore x·y = br+t
, which in log-form is
logb(x·y) = r + t = logb(x)+logb(y).

16. 1. logb(1) = 0
2. logb(x·y) = logb(x)+logb(y)
3. logb( ) = logb(x) – logb(y)
4. logb(xt
) = t·logb(x)
x
y
1. b0
= 1
2. br
· bt
= br+t
3. = br-t
4. (br
)t
= brt
bt
br
Properties of Logarithm
Recall the following
Rules of Exponents:
The corresponding
Rules of Logs are:
We veryify part 2: logb(xy) = logb(x) + logb(y), x, y > 0.
Proof:
Let x and y be two positive numbers. Let logb(x) = r
and logb(y) = t, which in exp-form are x = br
and y = bt
.
Therefore x·y = br+t
, which in log-form is
logb(x·y) = r + t = logb(x)+logb(y).The other rules may be verified similarly.

17. Example A:
3x2
√y
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).

18. 3x2
√y
log( ) = log( ),3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
Example A:

19. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
Example A:

20. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule
= log(3) + log(x2
)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
Example A:

21. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule power rule
= log(3) + log(x2
) – ½ log(y)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
Example A:

22. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule power rule
= log(3) + log(x2
) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
Example A:

23. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule power rule
= log(3) + log(x2
) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
Example A:

24. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule power rule
= log(3) + log(x2
) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2
) – log(y1/2
)
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
Example A:

25. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule power rule
= log(3) + log(x2
) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2
) – log(y1/2
) product rule
= log (3x2
) – log(y1/2
)
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
Example A:

26. 3x2
√y
log( ) = log( ), by the quotient rule
= log (3x2
) – log(y1/2
)
product rule power rule
= log(3) + log(x2
) – ½ log(y)
= log(3) + 2log(x) – ½ log(y)
3x2
√y
3x2
y1/2
Properties of Logarithm
a. Write log( ) in terms of log(x) and log(y).
log(3) + 2log(x) – ½ log(y) power rule
= log(3) + log(x2
) – log(y1/2
) product rule
= log (3x2
) – log(y1/2
)= log( )3x2
y1/2
b. Combine log(3) + 2log(x) – ½ log(y) into one log.
Example A:

27. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Properties of Logarithm

28. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
Properties of Logarithm

29. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
we have that:
a. logb(expb(x)) = x or logb(bx
) = x
Properties of Logarithm

30. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
we have that:
a. logb(expb(x)) = x or logb(bx
) = x
b. expb(logb(x)) = x or blog (x)
= x
Properties of Logarithm
b

31. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
we have that:
a. logb(expb(x)) = x or logb(bx
) = x
b. expb(logb(x)) = x or blog (x)
= x
Properties of Logarithm
b
Example B: Simplify
a. log2(2-5
) =
b. 8log (xy)
=
c. e2+ln(7)
=
8

32. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
we have that:
a. logb(expb(x)) = x or logb(bx
) = x
b. expb(logb(x)) = x or blog (x)
= x
Properties of Logarithm
b
Example B: Simplify
a. log2(2-5
) = -5
b. 8log (xy)
=
c. e2+ln(7)
=
8

33. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
we have that:
a. logb(expb(x)) = x or logb(bx
) = x
b. expb(logb(x)) = x or blog (x)
= x
Properties of Logarithm
b
Example B: Simplify
a. log2(2-5
) = -5
b. 8log (xy)
= xy
c. e2+ln(7)
=
8

34. Recall that given a pair of inverse functions, f and f -1
,
then f(f -1
(x)) = x and f -1
(f(x)) = x.
Since expb(x) and logb(x) is a pair of inverse functions,
we have that:
a. logb(expb(x)) = x or logb(bx
) = x
b. expb(logb(x)) = x or blog (x)
= x
Properties of Logarithm
b
Example B: Simplify
a. log2(2-5
) = -5
b. 8log (xy)
= xy
c. e2+ln(7)
= e2
·eln(7)
= 7e2
8

35. Ex. A
Disassemble the following log expressions in terms of
sums and differences logs as much as possible.
Properties of Logarithm
5. log2(8/x4
) 6. log (√10xy)
y√z3
2. log (x2
y3
z4
)
4. log ( )
x2
1. log (xyz)
7. log (10(x + y)2
) 8. ln ( )
√t
e2
9. ln ( )
√e
t2
10. log (x2
– xy) 11. log (x2
– y2
) 12. ln (ex+y
)
13. log (1/10y
) 14. log ( )x2
– y2
x2
+ y2 15. log (√100y2
)
3
3. log ( )
z4
x2
y3

36. Ex. B
Assemble the following log expressions into one log.
Properties of Logarithm
17. log(x) – log(y) + log(z) – log(w)
18. –log(x) + 2log(y) – 3log(z) + 4log(w)
19. –1/2 log(x) –1/3 log(y) + 1/4 log(z) – 1/5 log(w)
16. log(x) + log(y) + log(z) + log(w)
21. –1/2 log(x – 3y) – 1/4 log(z + 5w)
20. log(x + y) + log(z + w)
22. ½ ln(x) – ln(y) + ln(x + y)
23. – ln(x) + 2 ln(y) + ½ ln(x – y)
24. 1 – ln(x) + 2 ln(y)
25. ½ – 2ln(x) + 1/3 ln(y) – ln(x + y)

37. Properties of Logarithm
32. 101+log(5)
Ex. C
Simplify the following expressions.
30. 10log (x+y)
26. log2(2t
) 27. log(10x+y
)
29. 2log (t)2
28. ln(e√e
)
31. eln(√e)
33. eln(5) –1