Properties of logarithms

1. Properties
of
Logarithms

2. Since logs and exponentials of the same base are
inverse functions of each other they “undo” each other.
() () xxfaxf
a
x
log
1
==
-
Remember that: xffxff ==
--

11
and
This means that:

log1
xaff
x
a
==
-

xaff
x
a==
-
log
1

5log
2
2
inverses
“undo” each
each other
= 5
7
3
3log= 7

3. 3.
2.
1.
CONDENSED EXPANDED
Properties of
Logarithms
NM
aa
loglog+
=
=
=
NM
aa
loglog-
= Mr
a
log
(these properties are based on rules of
exponents since logs = exponents)
( )MN
alog
N
M
alog
r
a
Mlog

4. Using the log properties, write the expression as a
sum and/or difference of logs (expand).
÷
÷
ø
ö
ç
ç
è
æ
32
4
6
log
c
ab
using the second property:
3
2
6
4
6 loglog cab-
When working with logs, re-write any radicals as rational
exponents.
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
=
3
2
4
6log
c
ab
using the first property:
3
2
6
4
66 logloglog cba -+
using the third property: cba
666 log
3
2
log4log -+
NM
N
M
aaa logloglog -=
NMMN
aaa
logloglog +=
MrM
a
r
a loglog =

5. Using the log properties, write the expression as a
single logarithm (condense).
yx
33 log
2
1
log2 -
using the third property:
using the second property:
2
1
2
3
log
y
x
2
1
3
2
3
loglog yx-
MrM
a
r
a loglog =
NM
N
M
aaa logloglog -=
this direction
this direction

6. Use log
5
3≈.683 and log
5
7≈1.209
•Approximate:
•log
5
3/7 =
•log
5
3 – log
5
7 ≈
•.683 – 1.209 =
•-.526
•log
5
21 =
•log
5
(3·7)=
•log
5
3 + log
5
7≈
•.683 + 1.209 =
•1.892

7. Use log
5
3≈.683 and log
5
7≈1.209
•Approximate:
•log
5
49 =
•log
5
7
2
=
•2 log
5
7 ≈
•2(1.209)=
•2.418

8. Expanding Logarithms
•You can use the properties to expand logarithms.
•log
2 =
•log
2
7x
3
- log
2
y =
•log
27 + log
2x
3
– log
2y =
•log
2
7 + 3·log
2
x – log
2
y
y
x
3
7

9. Your turn!
•Expand:
•log 5mn
=
•log 5 + log

m + log

n
•Expand:
•log
5
8x
3
=
•log
5
8 + 3·log
5
x

10. Condensing Logarithms
•log 6 + 2 log2 – log 3 =
•log 6 + log 2
2
– log 3 =
•log (6·2
2
) – log 3 =
•log =
•log 8
3
26
2
×

11. Write the following expression as a single logarithm.

12. Your turn again!
•Condense:
•log
5
7 + 3·log
5
t =
•log
5
7t
3
•Condense:
•3log
2x – (log
24 + log
2y)=
•log
2
y
x
4
3

13. More Properties of
Logarithms
NMNM
aa
loglog then , If ==
NMNM
aa
== then ,loglog If
This one says if you have an equation, you can take
the log of both sides and the equality still holds.
This one says if you have an equation and each side
has a log of the same base, you know the "stuff" you
are taking the logs of are equal.

14. =8log
2
(2 to the what is 8?)
3
=16log
2
(2 to the what is 16?)
4
=10log
2
(2 to the what is 10?)
There is an answer to this and it must
be more than 3 but less than 4, but
we can't do this one in our head.
x=10log
2
Let's put it equal to x and we'll solve for x.
Change to
exponential form.
102=
x
NMNM
aa
loglog then , If ==
use log property & take log of
both sides (we'll use common log)
10log2log=
x
use 3rd log property
MrM
a
r
a
loglog =
10log2log=x
solve for x by
dividing by log 2
2log
10log
=x 32.3»
use calculator to
approximate
32.3»
Check by
putting 2
3.32
in
your calculator
(we rounded so
it won't be
exact)

15. Common and Natural Logarithms

16. Change of Base Formula
•The 2 bases we are most able to calculate logarithms
for are base 10 and base e. These are the only bases
that our calculators have buttons for.
•For ease of computing a logarithm, we may want to
switch from one base to another.
•The new base, a, can be any integer>1, but we often let
a=10 or a=e. (We know how to calculate common logs
and natural logs!)
b
M
M
a
a
b
log
log
log=

17. Common Logarithms
•A common logarithm has a base of 10.
•If there is no base given explicitly, it is
common.
•You can easily find common logs of powers
of ten.
•You can use your calculator to evaluate
common logs.

18. Change of base formula:
•u, b, and c are positive numbers with b≠1 and c≠1. Then:
•log
c
u =
•log
c
u = (base 10)
•log
c
u = (base e)
c
u
b
b
log
log
c
u
log
log
c
u
ln
ln

19. Compute
•What is the log, base 5, of 29?
•Does this answer make sense? What power would you
raise 5 to, to get 29? A little more than 2! (5 squared
is 25, so we would expect the answer to be slightly
more than 2.)
09.2
61.1
37.3
)5ln(
)29ln(
)29(log
5 ===

20. Use the Change-of-Base Formula and a calculator to
approximate the logarithm. Round your answer to three
decimal places.
16log
3
Since 3
2
= 9 and 3
3
= 27, our answer of what exponent to
put on 3 to get it to equal 16 will be something between
2 and 3.
3ln
16ln
16log
3
=
put in calculator
524.2»

21. Example
Find log
6
8 using common logarithms.
Solution: First, we let a = 10, b = 6, and M = 8.
Then we substitute into the change-of-base
formula:
10
1
6
0
log
log
l
8
og
1.1606
6
8=
»
log
log .
log
a
b
a
M
M
b
=

22. Example
•We can also use base e for a conversion.
Find log
6
8 using natural logarithms.
Solution: Substituting e for a, 6 for b and 8
for M, we have
6
6
log
log
log
ln8
1.1606
ln6
8
8
e
e
=
= »

23. Examples:
•Use the change of base to evaluate:
•log
3
7 =
•(base 10)
•log 7 ≈
•log 3
•1.771
•(base e)
•ln 7 ≈
•ln 3
•1.771

24. Solving equations
•Use the properties we have learned about exponential &
logarithmic expressions to solve equations that have
these expressions in them.
•Find values of x that will make the logarithmic or
exponential equation true.
•For exponential equations, if the base is the same on
both sides of the equation, the exponents must also be
the same (equal!)
NMbb
NM
==,

25. Sometimes it is easier to solve a logarithmic
equation than an exponential one
•Any exponential equation can be rewritten as a logarithmic
one, then you can apply the properties of logarithms
•Example: Solve:
93.1
12055.2
12
)5ln(
)99ln(
12)99(log
995
5
12
»
-=
-=
-=
=
-
x
x
x
x
x

26. SOLVE
56.4
54.46.2
1246.146.1
12
3ln
5ln
)1(
125log)1(
12)5(log
53
3
1
3
112
»
=
-=+
-=×+
-=×+
-=
=
+
+-
x
x
xx
xx
xx
x
x
xx

27. SOLVE
21.
319
67
0
15
67319
0
15
643203
0
15
)15(64
15
3
64
15
3
4
3
15
3
log
3)15(log)3(log
3
4
44
»=
=
-
+-
=
-
+-+
=
-
-
-
-
+
=
-
+
=
=÷
ø
ö
ç
è
æ
-
+
=--+
x
x
x
x
xx
x
x
x
x
x
x
x
x
xx

28. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au

29. Change-of-Base Formula
The base you change to can
be any base so generally
we’ll want to change to a
base so we can use our
calculator. That would be
either base 10 or base e.
LOG
“common”
log base 10
LN
“natural” log
base e
a
M
log
log
=
a
M
ln
ln
=
Example
for TI-83
If we generalize the process we just did
we come up with the:
a
M
M
b
b
a
log
log
log=