This document discusses the properties and laws of logarithms. It explains that common and natural logarithms have the same properties and laws, even though their bases differ. The key properties covered include: logarithms are only defined for positive values; ln1=0, ln e=1; and the product, quotient, and power laws. Several examples are worked through to demonstrate how to use these properties and laws to simplify logarithmic expressions and solve equations. The document emphasizes that understanding logarithmic properties and how to manipulate logarithms is important for advanced math, science, and engineering courses.

1. 5.5: Properties & Laws of
Logarithms
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Use properties and laws
of logarithms to simplify
and evaluate expressions.

2. Important Idea
The definitions of common
and natural logarithms
differ only in their bases,
therefore, they share the
same properties and laws.

3. Important Idea
Properties of Common
Logarithms:
•log v defined only for v>0
•log 1=0 & log 10=1
• log10k
k=
• for v>0log
10 v
v=

4. Important Idea
Properties of Natural
Logarithms:
•ln v defined only for v>0
•ln 1=0 & ln e=1
• ln k
e k=
• for v>0
lnv
e v=

5. Example
Use the properties of
logarithms to solve the
equation:
ln( 1) 2x
e e+
=
log( 2) 2x − =
ln( 4) 2x + = −

6. Try this
Use the properties of
logarithms to solve the
equation:
log( 3) 1x − = x=13
3 5.718x e= + ≈ln( 3) 1x − =

7. Important Idea
ln(ab)=
ln a + ln b
ln an
=n ln a
ln ln ln
a
a b
b
 
= − 
 
Product Law:
Quotient Law:
Power Law:

8. Example
log33Find given log3 .4771=
and log11 1.0414=
What law was used?

9. Example
ln63Find given ln7 1.9459=
and ln9 2.1972=
What law was used?

10. Try This
find given
and
log12 log6 .7782=
log2 .3010=
log12 1.0792=
Using the product law,

11. Example
Using the product law,
write the given expression
as a single logarithm:
2
ln lnx x+
log(2 ) log( 1)x x+ +

12. Try This
Using the product law,
write the given expression
as a single logarithm:
ln( 1) ln( 1)x x+ + −
2
ln( 1)x −

13. Try This
Using the product law,
write the given expression
as two logarithms:
2
ln( 2)x x+ −
ln( 2) ln( 1)x x+ + −

14. Example
log3Find given log12 1.0792=
and log4 .6021=
What law was used?

15. Try This
log3Find given log6 .7782=
and log2 .3010=
log3 .4771=

16. Example
Using the quotient law,
write the given expression
as a single logarithm:
2
ln lnx x−
log(2 ) log( 1)x x− +

17. Try This
Using the quotient law, write
the given expression as a
single logarithm:
ln( 1) ln( 1)x x+ − −
1
ln
1
x
x
+ 
 ÷
− 

18. Try This
Using the quotient law, write
the given expression as two
logarithms:
ln( 1) ln( 1)x x+ − −
1
ln
1
x
x
+ 
 ÷
− 

19. Example
Using the power law, re-
write the given expression
and simplify if possible:
2
ln x 3log(2 )x
( 1)
ln( 1) x
x +
+ ( 1)ln( 1)x x+ +

20. Example
Using the power law, re-
write the given expression
and simplify if possible:
2
ln x 3log(2 )x
( 1)
ln( 1) x
x +
+ ( 1)ln( 1)x x+ +

21. Try This
Using the power law, re-
write the given expression
and simplify if possible:
2
log4
3
lne
5
log10
2log4
3ln 3e =
5log10 5=

22. Example
Use a combination of
logarithmic properties and
laws to re-write the given
expression:
2
2( 3)
ln
1
x
x
 +
 ÷
− 

23. Example
Use a combination of
logarithmic properties and
laws to re-write the given
expression:
3
10
log
1
x
x
 
 ÷
+ 

24. Try This
Use a
combination of
logarithmic
properties and
laws to re-
write the given
expression:
3
( 5)
ln
1
e x
x
 −
 ÷
+ 
1 3ln( 5) ln( 1)x x+ − − +

25. Example
The 1989 world series
earthquake in San Francisco
measure 7.0 on the Richter
Scale. The great earthquake
of 1906 measured 8.3. How
much more intense was the
1906 quake?
( )0logR i i=

26. Example
Decibels are calculated by
the function where
is the minimum sound
intensity detectable by the
human ear. Find the decibel
level of a jet engine which is
10 billion times
010log( )i i
0i
0.i

27. Lesson Close
The manipulation of
logarithms is a fundamental
math skill that you will need
in upper level math courses
and in science and
engineering.