College Algebra. Logarithms: Denitions and Domains. Dr. Nguyen November 9, Department of Mathematics UK

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1 College Algebra Logarithms: Denitions and Domains Dr. Nguyen Department of Mathematics UK November 9, 2018

2 Agenda Logarithms and exponents Domains of logarithm functions

3 Operations Each arithmetic operation has its inverse/opposite: Addition x + a Subtraction x a Multiplication x a Division x/a ath Power x a ath Root a x Exponentiation a x Logarithm log a (x) base a base a

4 The Opposite of Exponents: Logarithms Let x and y be real numbers with x > 0, and a be a positive number not equal to 1. Then log a (x) (log base a) is a number y for which a y = x. That is, log a (x) = y means a y = x. Abbreviations: log 10 (x) is abbreviated log(x). log e (x) is abbreviated ln(x).

5 The Opposite of Exponents: Logarithms Let x and y be real numbers with x > 0, and a be a positive number not equal to 1. Then log a (x) (log base a) is a number y for which a y = x. That is, log a (x) = y means a y = x. Abbreviations: log 10 (x) is abbreviated log(x). log e (x) is abbreviated ln(x).

6 Operations Each arithmetic operation has its inverse/opposite: Addition x + a Subtraction x a Multiplication x a Division x/a ath Power x a ath Root a x Exponentiation a x Logarithm log a (x) base a base a

7 C1: Calculator Example On a TI-84 calculator, the log button is the logarithm base 10. Try calculating log 10 (3) = log(3) = Type a number (to at least four decimal places).

8 C2: Calculator Example On a TI-84 calculator, the log button is the logarithm base 10. Try calculating log 10 (3) = log(3) = log(3) Then take 10, and raise it to your result: = What did you get? Type a number (to at least four decimal places).

9 Calculator Example On a TI-84 calculator, the log button is the logarithm base 10. log(3) Notice that , the input of the logarithm. This is not a coincidence: log(3) is the number (exponent) to which 10 must be raised to produce 3.

10 C3: Calculator Example On a TI-84 calculator, the log button is the logarithm base 10. Try calculating Type a number. log(10 2 ) =

11 C4: Calculator Example On a TI-84 calculator, the log button is the logarithm base 10. Try calculating log(10 2 ) = log(10 2 ) = log(100) = 2. Then take 10, and raise it to your result: 10 2 =... What did you get? Type a number.

12 Calculator Example On a TI-84 calculator, the log button is the logarithm base 10. log(10 2 ) = 2. Notice that 10 2 = 10 2, the input of the logarithm. This is not a coincidence: log(10 2 ) is the number (exponent) to which 10 must be raised to produce 10 2, or 100. Notice we can read o the answer from the exponent on 2 in the input.

13 Logarithm of an Exponent In general, for any base a, For something like log a (a EXPONENT ) = EXPONENT log 10 (10 2 ) = 2, we can just read o the answer from the input's exponent, but if the input is not written as a power of the base, log 10 (100), we have to write the input as a power of the base: 100 = 10 2, so log 10 (100) = log 10 ( 10 2 ) = 2.

14 Logarithm of an Exponent In general, for any base a, For something like log a (a EXPONENT ) = EXPONENT log 10 (10 2 ) = 2, we can just read o the answer from the input's exponent, but if the input is not written as a power of the base, log 10 (100), we have to write the input as a power of the base: 100 = 10 2, so log 10 (100) = log 10 ( 10 2 ) = 2.

15 Logarithm of an Exponent In general, for any base a, For something like log a (a EXPONENT ) = EXPONENT log 10 (10 2 ) = 2, we can just read o the answer from the input's exponent, but if the input is not written as a power of the base, log 10 (100), we have to write the input as a power of the base: 100 = 10 2, so log 10 (100) = log 10 ( 10 2 ) = 2.

16 In general, for any base a, For something like log a (a EXPONENT ) = EXPONENT log 10 (10 2 ) = 2, we can just read o the answer from the input's exponent, but if the input is not written as a power of the base, log 10 (100), we have to write the input as a power of the base: Meanwhile, for we really have 100 = 10 2, so log 10 (100) = log 10 ( 10 2 ) = 2. log 10 (3), , so log 10 (3) log 10 ( ) =

17 In general, for any base a, For something like log a (a EXPONENT ) = EXPONENT log 10 (10 2 ) = 2, we can just read o the answer from the input's exponent, but if the input is not written as a power of the base, log 10 (100), we have to write the input as a power of the base: Meanwhile, for we really have 100 = 10 2, so log 10 (100) = log 10 ( 10 2 ) = 2. log 10 (3), , so log 10 (3) log 10 ( ) =

18 Ex1: Whole Number Logarithm Compute log 3 (81). That is, 3 raised to what number equals 81?

19 Ex1: Whole Number Logarithm Compute log 3 (81). That is, 3 raised to what number equals 81? Some powers of 3: Since 3 4 = 81, this means 3 1 = = = = 81 log 3 (81) = 4.

20 Ex2: Negative Number Logarithm Compute log 3 ( 1 9 ). That is, 3 raised to what number equals 1 9?

21 Ex2: Negative Number Logarithm Compute log 3 ( 1 9 ). That is, 3 raised to what number equals 1 9? Looks like reciprocals are involved. Try negative exponents: 3 1 = = 1 9 Since 3 2 = 1, this means 9 ( ) 1 log 3 = 2. 9

22 Ex3: Negative Number Logarithm Compute log 1/2 (8). That is, 1 raised to what number equals 8? 2

23 Ex3: Negative Number Logarithm Compute log 1/2 (8). That is, 1 raised to what number equals 8? 2 Looks like reciprocals are involved. Try negative exponents: ( ) 1 1 = 1 2 1/2 = 2 ( ) 2 ( ) = = (2) 2 = 4 2 1/2 ( ) 3 ( ) = = (2) 3 = 8 2 1/2

24 Ex3: Negative Number Logarithm Compute log 1/2 (8). That is, 1 raised to what number equals 8? ( 2) 3 1 Since = 8, this means 2 log 1/2 (8) = 3.

25 Ex4: Fraction Logarithm Compute log 49 (7). That is, 49 raised to what number equals 7?

26 Ex4: Fraction Logarithm Compute log 49 (7). That is, 49 raised to what number equals 7? No reciprocals, but roots are involved: 49 = 7 Remember that square roots are the same as 1/2 powers: Since 49 1/2 = 7, this means 49 (1/2) = 7 log 49 (7) = 1 2.

27 C5: Compute Logarithm What number is Don't use a calculator. log 10 ( )?

28 C5: Compute Logarithm log 10 ( ) = 95.1 log 10 ( ) is the number to which 10 is raised to produce an output of However, the input is already written as a power of 10, so we can just read o the answer from the exponent on 10.

29 C6: Compute Logarithm What number is log 4 (64)? Hint: 4 to what number equals 64?

30 C6: Compute Logarithm log 4 (64) = 3 log 4 (64) is the number to which 4 is raised to produce an output of 64. We have 4 3 = 64, so log 4 (64) = log 4 (4 3 ) = 3.

31 Some Base 4 Logarithms 4 x = y log 4 (y) = x 4 0 = 1 log 4 (1) = = 4 log 4 (4) = = 16 log 4 (16) = = 64 log 4 (64) = 3 4 1/2 = 4 = 2 log 4 (2) = 1/2 4 1 = 1/4 log 4 (1/4) = 1

32 Some Base 4 Logarithms 4 x = y 4 3/2 = 4 3 = 64 = 8 log 4 (y) = log 4 (4 x ) = x log 4 (8) = log 4 (4 3/2 ) = 3/2 4 2 = 1/(4 2 ) = 1/16 log 4 (1/16) = log 4 (4 2 ) = 2 4 1/2 = 1/(4 1/2 ) = 1/2 4 1/3 = 3 4 log 4 (1/2) = log 4 (4 1/2 ) = 1/2 log 4 ( 3 4) = log 4 (4 1/3 ) = 1/3

33 Some Base e Logarithms (Extra) e x = y ln(y) = ln(e x ) = x e 0 = 1 ln(1) = ln(e 0 ) = 0 e 1 = e ln(e) = ln(e 1 ) = 1 e ln(7.3891) ln(e 2 ) = 2 e ln( ) ln(e 3 ) = 3 It is better to keep expressions like e 2 as e 2 instead of using a decimal approximation.

34 Domain of a Logarithm Function For any base a, let f (x) = log a (INPUT ), where the INPUT could be x, or an expression in x. Then the domain of f is the set of solutions to INPUT > 0.

35 Ex5: Domain (Simple Input) What is the domain of f (x) = ln(x)?

36 Ex5: Domain (Simple Input) What is the domain of f (x) = ln(x)? The domain of f is the set of solutions to x > 0, or the interval (0, ).

37 Ex6: Domain (Linear Input) What is the domain of g(x) = log 8 ( x 5)?

38 Ex6: Domain (Linear Input) What is the domain of g(x) = log 8 ( x 5)? The domain of g is the set of solutions to x 5 > 0 x > 5 x < 5 which is the interval (, 5).

39 Next Time Please read Section 6.3 of your textbook. We will do some more examples of logarithm domain problems, and review domain, function composition, and inverses as well.