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5.4 - Common Logarithms

2. 4/20/2019 8:41 AM
5.4 - Common Logarithms

3. Logarithmic Functions
Section 5.4
Pre-Calculus AB PreAP, Revised ©2015
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5.4 - Common Logarithms

4. 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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5.4 - Common Logarithms

5. 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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5.4 - 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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5.4 - 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.
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5.4 - Common Logarithms

8. 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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5.4 - Common Logarithms

9. Review What is the inverse function of y = 2x? 4/20/2019 8:41 AM
5.4 - Common Logarithms

10. To Identify Logarithms
b = Base
x = Power/Argument
a = Value
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11. The Snail
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12. Example 1
Given 24 = 16, write this problem in logarithmic form
Identify the components of this problem
EXPONENT
ARGUMENT
BASE
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13. = Example 1 Given 24 = 16, write this problem in logarithmic form
5.4 - Common Logarithms

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

15. Example 3
Given log4(1/16) = –2, write this problem in exponential form
Identify the components of this problem
ARGUMENT
EXPONENT
BASE
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5.4 - Common Logarithms

16. Example 3
Given log4(1/16) = –2, write this problem in exponential form =
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5.4 - Common Logarithms

17. Your Turn Given 10–2 = 1/100, write this problem in logarithmic form
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5.4 - Common Logarithms

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

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

20. Example 5
Given log279 = x, write this problem in exponential form and solve for x (without a calculator)
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21. Example 6
Given log832 = x, write this problem in exponential form and solve for x (without a calculator)
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5.4 - Common Logarithms

22. Your Turn
Given log 𝟖 𝟖 , write this problem in exponential form and solve (without a calculator)
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23. Example 7
Given log 100 = x, write this problem in exponential form and solve for x
If there isn’t a base given, assume the base to be…
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5.4 - Common Logarithms

24. Your Turn
Given log 1/1000 = x, write this problem in exponential form and solve for x
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25. Assignment
Worksheet
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5.4 - Common Logarithms