1. Properties of Logarithms Lesson 7-4
Algebra 2
Properties of Logarithms
Lesson 7-4

2. Goals Goal Rubric To use the properties of logarithms.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Change of Base Formula

4. Essential Question Big Idea: Modeling
How are the properties of logarithms similar to the properties of exponents?

5. History of Logarithms
John Napier, a 16th Century Scottish scholar, contributed a host of mathematical discoveries.
He is credited with creating the first computing machine, logarithms and was the first to describe the systematic use of the decimal point.
Napier lived during a time when revolutionary astronomical discoveries were being made.
But 16th century arithmetic was barely up to the task and Napier became interested in this problem.
JOHN NAPIER ( ) Born at Merchiston, Edinburgh, Napier was a man of great intellect, persistence and will, known best for his development of logarithms in the field of mathematics. He had diverse interests outside his solitary and studious life, including treasure hunting, agricultural chemistry, and divination. He was also responsible for inventing the world's first computing device, nicknamed 'Napier's Bones' on account of its design. Napier's bones were multiplication tables written on strips of wood or bones. The invention was used for multiplying, dividing, and taking square roots and cube roots.
'Constructio', which described the means by which he constructed the log tables, was published after his death.
Napier is famous for creating mathematical logarithms, creating the decimal point, and for inventing Napier's Bones, a calculating instrument. John Napier - Inventor
He proposed several military inventions including: burning mirrors that set enemy ships on fire, special artillery that destroyed everything within a radius of four miles, bulletproof clothing, a crude version of a tank, and a submarine-like device. John Napier invented a hydraulic screw with a revolving axle that lowered water levels in coal pits.
The highlight of John Napier's life was the creation of logarithms and the decimal notation for fractions. His other mathematical contributions included: a mnemonic for formulas used in solving spherical triangles, two formulas known as Napier's analogies used in solving spherical triangles, and the exponential expressions for trigonometric functions.
John Napier (1550 – 1617)

6. Napier’s Bones
In 1617, the last year of his life, Napier invented a tool called “Napier's Bones” which reduces the effort it takes to multiply numbers.
Napier's Bones were invented in 1617 , when John Napier, a Scottish baron, published a book describing the device. Within a few years, it had spread throughout Europe and as far as China. Napier's Bones (so-called because they were often made of bone) were rods with multiplications tables on them. At the time, educated people often knew their multiplication tables only as far as 5 x 5.
“Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”

7. Logarithms Appear
The first definition of the logarithm was constructed by Napier and popularized by a pamphlet published in 1614, two years before his death. His goal: reduce multiplication, division, and root extraction to simple addition and subtraction.
Napier defined the "logarithm" L of a number N by:
N==107(1-10(-7))L
This is written as NapLog(N) = L or NL(N) = L
Numerical calculation
The development of new methods of numerical calculation was a response to the increased practical demands of numerical computation, particularly in trigonometry, navigation, and astronomy. New ideas spread quickly across Europe and resulted by 1630 in a major revolution in numerical practice.
Simon Stevin of Holland, in his short pamphlet La Disme (1585), introduced decimal fractions to Europe and showed how to extend the principles of Hindu-Arabic arithmetic to calculation with these numbers. Stevin emphasized the utility of decimal arithmetic "for all accounts that are encountered in the affairs of men," and he explained in an appendix how it could be applied to surveying, stereometry, astronomy, and mensuration. His idea was to extend the base-10 positional principle to numbers with fractional parts, with a corresponding extension of notation to cover these cases. In his system the number was denoted
in which the digits to the left of the zero are the integral part of the number. To the right of the zero are the digits of the fractional part, with each digit succeeded by a circled number that indicates the negative power to which 10 is raised. Stevin showed how the usual arithmetic of whole numbers could be extended to decimal fractions using rules that determined the positioning of the negative powers of 10.
Tables of logarithms were first published in 1614 by the Scottish baron John Napier in his treatise Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms). This work was followed (posthumously) five years later by another in which Napier set forth the principles used in the construction of his tables. The basic idea behind logarithms is that addition and subtraction are easier to perform than multiplication and division, which, as Napier observed, require a "tedious expenditure of time" and are subject to "slippery errors." By the law of exponents, anam = an + m, that is, in the multiplication of numbers the exponents are related additively. By correlating the geometric sequence of numbers a, a2, a3, (a is called the base) and the arithmetic sequence 1, 2, 3, and interpolating to fractional values, it is possible to reduce the problem of multiplication and division to one of addition and subtraction. To do this Napier chose a base that was very close to 1, differing from it by only 1/107. The resulting geometric sequence therefore yielded a dense set of values, suitable for constructing a table.
While Napier's definition for logarithms is different from the
modern one, it transforms multiplication and division into
addition and subtraction in exactly the same way.

8. Properties of Logarithms
Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

9. Product Property
Remember that to multiply powers with the same base, you add exponents.

10. Product Property Think: log j + log a + log m = log jam
The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified.
Helpful Hint
Think: log j + log a + log m = log jam

11. Example: Express log64 + log69 as a single logarithm. Simplify.
To add the logarithms, multiply the numbers.
log6 (4  9)
log6 36
Simplify. 2
Think: 6? = 36.

12. Your Turn: Express as a single logarithm. Simplify, if possible.
log log525
log5 (625 • 25)
To add the logarithms, multiply the numbers.
log5 15,625
Simplify. 6
Think: 5? = 15625

13. Your Turn: Express as a single logarithm. Simplify, if possible.
log log 1 3 9 1 3
log ( 27 • ) 9
To add the logarithms, multiply the numbers. 1 3
log 3
Simplify. –1
Think: ? = 3 1 3

14. Quotient Property
Remember that to divide powers with the same base, you subtract exponents
Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

15. Quotient Property
The property on the last slide can also be used in reverse.
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution

16. Example:
Express log5100 – log54 as a single logarithm. Simplify, if possible.
log5100 – log54
To subtract the logarithms, divide the numbers.
log5(100 ÷ 4)
log525
Simplify. 2
Think: 5? = 25.

17. Your Turn:
Express log749 – log77 as a single logarithm. Simplify, if possible.
log749 – log77
To subtract the logarithms, divide the numbers
log7(49 ÷ 7)
log77
Simplify. 1
Think: 7? = 7.

18. Power Property
Because you can multiply logarithms, you can also take powers of logarithms.

19. Example: Express as a product. Simplify, if possible. A. log2326
B. log8420
20log84
6log232
Because = 32, log232 = 5.
Because = 4, log84 = 2 3
6(5) = 30
20( ) = 40 3 2

20. Your Turn: Express as a product. Simplify, if possibly. a. log104
b. log5252
4log10
2log525
Because = 10, log 10 = 1.
Because = 25, log525 = 2.
4(1) = 4
2(2) = 4

21. Your Turn: Express as a product. Simplify, if possibly. c. log2 ( )51 2
5log2 ( ) 1 2
Because –1 = , log = –1. 1 2
5(–1) = –5

22. Summary Properties of Logs
Product Rule
Quotient Rule
Power Rule
The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra.
This is because the properties convert more complicated products, quotients, and exponential forms into simpler sums, differences, and products.
This is called expanding a logarithmic expression.
The procedure above can be reversed to produce a single logarithmic expression.
This is called condensing a logarithmic expression.

23. log 5mn = log 5 + log m + log n log58x3 = log58 + 3·log5x Examples:
Expand:
log 5mn =
log 5 + log m + log n
log58x3 =
log58 + 3·log5x

24. Expand: Express as a Sum and Difference of Logarithms
Your Turn:
log2 =
log27x3 - log2y =
log27 + log2x3 – log2y =
log27 + 3·log2x – log2y
Expand: Express as a Sum and Difference of Logarithms

25. Condense - Express as a Single Logarithm
Example: Write the following as the logarithm of a single expression.
Power Rule
Product Rule
Quotient Rule

26. Condensing Logarithms
log log2 – log 3 =
log 6 + log 22 – log 3 =
log (6·22) – log 3 =
log =
log 8

27. Your Turn: log57 + 3·log5t = log57t3 3log2x – (log24 + log2y)= log2
Condense:
log57 + 3·log5t =
log57t3
3log2x – (log24 + log2y)=
log2

28. Your Turn: Express in terms of sums and differences of logarithms.
Solution:

29. Your Turn:
Express as a single logarithm.
Solution:

30. Another Type of Problem
If loga3 = x and loga4 = y, express each log expression in terms of x and y.
loga12
Loga(3•4) = loga3 + loga4 = x+y
Log34
Log34 = loga4/loga3 = y/x

31. Inverse Properties
Exponential and logarithmic operations undo each other since they are inverse operations.

32. Example: Simplify each expression. a. log3311 b. log381 c. 5log5104

33. Your Turn: a. Simplify log100.9 b. Simplify 2log2(8x) log 100.9

34. Change of Base
Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula.

35. Example: Evaluate log328. Method 1 Change to base 10 log8 log328 =
0.903
1.51 ≈
Use a calculator.
Divide.
≈ 0.6

36. Example: Continued Evaluate log328.
Method 2 Change to base 2, because both 32 and 8 are powers of 2.
log328 =
log28
log232 = 3 5
Use a calculator.
= 0.6

37. Your Turn: Evaluate log927. Method 1 Change to base 10. log27 log927 =
1.431
0.954 ≈
Use a calculator.
≈ 1.5
Divide.

38. Your Turn: Continued Evaluate log927.
Method 2 Change to base 3, because both 27 and 9 are powers of 3.
log927 =
log327
log39 = 3 2
Use a calculator.
= 1.5

39. Your Turn: Evaluate log816. Method 1 Change to base 10. log16 Log816 =
1.204
0.903 ≈
Use a calculator.
Divide.
≈ 1.3

40. Your Turn: Continued Evaluate log816.
Method 2 Change to base 4, because both 16 and 8 are powers of 2.
log816 =
log416
log48 = 2
1.5
Use a calculator.
= 1.3

41. Essential Question Big Idea: Modeling
How are the properties of logarithms similar to the properties of exponents?
The properties of logarithms are derived from the properties of exponents. Use the Product, Quotient, and Power Properties to condense or expand logarithms.

42. Essential Question Big Idea: Modeling
Why is the change of base formula useful?
The Change of Base Formula allows you to write a logarithmic expression in one base as an equivalent logarithmic expression in another base (usually base 10). Use the Change of Base Formula to evaluate logarithmic expressions with a calculator.

43. Assignment
Section 7-4, Pg 495 – 497; #1 – 8 all, 10 – 22 even, 26 – 38 even, 42.