Logarithms
I.
Logarithms
a. Definition: A logarithm is the exponent required to produce a
given number.
b. The General Form is: y  log a x . It can be rewritten as a y  x .
1.
Ex) log 2 8  3
In exponential form: 23  8
c. log a x is only defined if a and x are both positive, and a  1 .
d. The Common Logarithm has a base of 10. In general usage, the
subscript 10 is omitted when dealing with common logs.
1.
Ex) log10  1
In exponential form: 101  10
e. A Natural Logarithm is a logarithm where the base is the number “
e ” (where e  2.72828 ). log e x is replaced with ln x . Natural
logarithms play an important role in theoretical mathematics and
the natural sciences.
II.
Properties of Logs:
Identity
Example
Log of a Product:
log 2 24  log 2 6  log 2 4
log xy  log x  log y
Log of a Quotient:
log
x
 log x  log y
y
log 2
10
 log 2 10  log 2 3
3
 
Log with Exponent:
log 2 83  3 log 2 8
1
1
log 4 32  log32 4  log 32
4
log x a  a log x
1
1
log a x  log x a  log x
a
Identities:
log a a  1
log a 1  0
ln e  1
ln 1  0
log 3 3  1
log 2 1  0
Negative Exponents:
log a
1
  log a x
x
log 3
1
  log 3 9
9
Change in base (from base a to base 10):
log a x 
log 2 5 
log x ln x

log a ln a
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log 5 ln 5

log 2 ln 2
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III.
Solving Equations Using Logarithms
1. 4 x  7
Take the log of both sides. You can use the
ln 4 x  ln 7
common logarithm or natural log, but in
practice the natural log (or ln) is used more.
Use the exponent property to rewrite, then
divide to solve for x .
x ln 4  ln 7
x
ln 7
or log 4 7
ln 4
 
ln610   ln 11
2. 6 10 x  11x
x
Take the log of both sides.
x
ln 6  ln 10 x  ln 11x
ln 6  x ln 10  x ln 11
Use exponent property.
ln 6  x ln 11  x ln10
Get x terms on one side.
ln 6  xln 11  ln 10
Factor out the x terms.
ln 6
ln 11  ln 10
Divide and solve for x .
x
3. 4e x  7
ex 
7
4
Divide by 4 to get e x by itself.
ln e x  ln
7
4
Take the log of both sides.
x ln e  ln
7
4
Use exponent rule.
x(1)  ln
x  ln
7
4
Use identity that ln e  1 .
7
or x  ln 7  ln 4
4
Simplify.
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Practice Problems:
Write in exponential form:
1. log 3 27 =3
2. log 2 32 =5
3. log17 1 =0
Write it logarithmic form:
4. 6 2  36
5. 103  1000
6. 9
1
2
3
Solve for x:
7. log x 49  2
8. log 5 125  x
9. log 8 x  2
10. e x  11
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11.
1 x
e  20
4
12. 5 x  26
13. log 16 x 
1
2
14. log 7 x  1
15. log 3
1
x
9
Use the properties of logs to write in expanded form:
16. log 8 xz 
 u3 
17. log 7  4 
v 
 xy 2 
18. log 5  4 
 z 
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
19. ln x 2 yz
20. log 7

x3
y
Solve the following equations using logarithms:
21. 5 x  6
22. e x  3
23. 2 x1  6
24. 32 x1  4
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Answers to Logarithms:
1.
33  27
13.
x4
2.
25  32
14.
x
3.
17 0  1
15.
x  2
4.
log 6 36  2
16.
log 8 x  log 8 z
5.
log1000  3
17.
3 log 7 u  4 log 7 v
6.
log 9 3 
18.
log 5 x  2 log 5 y  4 log 5 z
7.
x7
19.
2 ln x  ln y  ln z
8.
x3
20.
3
1
log 7 x  log 7 y
2
2
9.
x  64
21.
x
10.
x  ln 11
22.
x  ln 3
11.
x  ln 80
23.
x
12.
x
ln 26
ln 5
24.
1  ln 4 
x 
 1
2  ln 3 
1
2
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1
7
ln 6
or log 5 6
ln 5
ln 6
1
ln 2
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