1. Worksheet Key
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7.4 - Common Logarithms

2. LOGARITHMIC FUNCTIONS
SECTION 7.4,
©2013,
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7.4 - Common Logarithms

3. Real-Life Situation
The pH scale is used in chemistry to determine the acidity or alkalinity of a solution.
The scale ranges from 1 to 14, with 1 being the most acidic and 14 the most alkaline
The difference in strength of an acid of pH 1 and that of pH 2 is not twofold, but tenfold
The pH scale is actually a logarithm in the form: log10, thus, pH 1 = log1010, and pH 2 = log10100
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7.4 - Common Logarithms

4. Real-Life Situation
The Richter scale is used to determine the strength of the ground movement. The larger the number, the more violent the movement
Similar to the pH scale, an earthquake of magnitude 7 on the scale is ten times stronger than an earthquake of magnitude 6.
Again, this is because the Richter scale is actually a logarithm: log10[measurement of movement of the earth]
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7.4 - Common Logarithms

5. Video
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7.4 - Common Logarithms

6. Real-Life Situation The Modified Richter Scale uses a modified scale.
It is not ten-fold
A separate equation is used
Music “Semitones”
The interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm).
Astronomy
The magnitude measures the stars’ brightness logarithmically with vision
Source: Wikipedia
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7.4 - Common Logarithms

7. Key Terms
Logarithms are defined as the INVERSE of an exponential function. It can be used specifically to find base powers. They are 10-fold.
Exponential form: bx = a; where b is the BASE, x is the POWER, and a is the value.
Logarithmic form: logba = x; b is the BASE, a is the ARGUMENT, and x is the value.
It is read as “log of a base b” or “log base b of a”
If the base is not given (such as log 3) it is understood to be COMMON BASE OF 10.
Negative bases for logarithms give valid “Laws of Logarithms” only in certain cases, i.e. where the arguments are even-numbered powers of the base
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7.4 - Common Logarithms

8. If there is not a base given, the base is ALWAYS 10.
Definition
Logarithms are defined as the INVERSE of an exponential function. It can be used specifically to find base powers. They are 10-fold.
If there is not a base given, the base is ALWAYS 10.
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7.4 - Common Logarithms

9. Quick Example What is the inverse function of y = 2x?
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7.4 - Common Logarithms

10. To Identify Logarithms
b = Base
x = Power/Argument
a = Value
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11. The Snail
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7.4 - Common Logarithms

12. Example 1
Given 24 = 16, write this problem in logarithmic form
Identify the components of this problem
POWER
VALUE
BASE
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7.4 - Common Logarithms

13. = Example 1 Given 24 = 16, write this problem in logarithmic form
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7.4 - Common Logarithms

14. Example 2 Given 43/2 = 8, write this problem in logarithmic form
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7.4 - Common Logarithms

15. Your Turn Given 10–2 = 1/100, write this problem in logarithmic form
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16. Example 3
Given log4(1/16) = –2, write this problem in exponential form
Identify the components of this problem
VALUE
ARGUMENT
BASE
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17. Example 3
Given log4(1/16) = –2, write this problem in exponential form =
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7.4 - Common Logarithms

18. Example 4
Given log7(1/49) = x, solve for x and write this problem in exponential form (without a calculator)
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7.4 - Common Logarithms

19. Example 5
Given log64(x) = 1/2, solve for x and write this problem in exponential form (without any calculator)
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20. Your Turn
Given logx64 = 2, solve for x and write this problem in exponential form (without a calculator)
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7.4 - Common Logarithms

21. Example 6
Given log279 = x, solve for x and write this problem in exponential form (without a calculator)
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7.4 - Common Logarithms

22. Example 7
Given log1664 = x, solve for x and write this problem in exponential form (without a calculator)
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7.4 - Common Logarithms

23. Your Turn
Given log832 = x, solve for x and write this problem in exponential form (without a calculator)
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24. Example 8
Given log 100 = x, solve for x and write this problem in exponential form
If there isn’t a base given, assume the base to be…
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25. Example 9
Given log = x, solve for x and write this problem in exponential form
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26. Your Turn
Given log 1/1000 = x, solve for x and write this problem in exponential form
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27. Parent Function Exponential Function: –2 –1 1 2 1 2 4 Domain: Range:
Process Column
f(x); y –2 –1 1 2 1 2 4
Domain:
Range:
Asymptote:
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7.4 - Common Logarithms

28. Parent Function Logarithmic Function: –2 –1 1 2 1 4 2 Domain: Range:x
Process Column
f(x); y –2 –1 1 2 1 2 4
Domain:
Range:
Asymptote:
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7.4 - Common Logarithms

29. Parent Function
What are the similarities and differences between these two graphs?
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7.4 - Common Logarithms

30. Example 10 Fill in the table, graph and state the domain and range –2
Process Column
f(x); y –2 –1 1 2 x
Process Column
f(x); y
1/9
1/3 1 3 9 –2 –1 1 2 1 3 9
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7.4 - Common Logarithms

31. Example 10 –2 –1 1 2 1/9 1/3 1 3 9 1/9 1/3 1 3 9 –2 –1 1 2 Domain:
f(x); y –2 –1 1 2 x
f(x); y
1/9
1/3 1 3 9
1/9
1/3 1 3 9 –2 –1 1 2
Domain:
Range:
Asymptote:
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7.4 - Common Logarithms

32. Your Turn Fill in the table, graph and state the domain and range x
y = (1/2)x y –3 –1 1 3
y = (1/2)–3
y = (1/2)–1
y = (1/2)0
y = (1/2)1
y = (1/2)3 8 2 1
1/8 x
y = log ½ (x) y –3 –1 1 3 8 2 1
1/8
–3 = log ½ (x); (1/2)–3 = x
–1 = log ½ (x); (1/2)–1 = x
0 = log ½ (x); (1/2)0 = x
1 = log ½ (x); (1/2)1 = x
3 = log ½ (x); (1/2)3 = x
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7.4 - Common Logarithms

33. Assignment
Worksheet
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7.4 - Common Logarithms