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Exploring Exponential and Logarithmic Functions

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When we are given the logarithm and asked to find the log, we are finding the antilogarithm. ... Find the antilogarithm of each. 2.498 c. -1.793. 0.164 d. 0.704 2 ... – PowerPoint PPT presentation

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Title: Exploring Exponential and Logarithmic Functions


1
Chapter 10
  • Exploring Exponential and Logarithmic Functions

By Kathryn Valle
2
10-1 Real Exponents and Exponential Functions
  • An exponential function is any equation in the
    form y abx where a ? 0, b gt 0, and b ? 1. b
    is referred to as the base.
  • Property of Equality for Exponential Functions
    If in the equation y abx, b is a positive
    number other than 1, then bx1 bx2 if and only
    if x1 x2.

3
10-1 Real Exponents and Exponential Functions
  • Product of Powers Property To simplify two like
    terms each with exponents and multiplied
    together, add the exponents.
  • Example 34 35 39
  • 5v2 5v7 5v2 v7
  • Power of a Power Property To simplify a term
    with an exponent and raised to another power,
    multiply the exponents.
  • Example (43)2 46
  • (8v5)4 84v5

4
10-1 Examples
  • Solve
  • 128 24n 1 53n 2 gt 625
  • 27 24n 1 53n 2 gt 54
  • 7 4n 1 3n 2 gt 4
  • 8 4n 3n gt 2
  • n 2 n gt ²/³

5
10-1 Practice
  • Simplify each expression
  • (23)6 c. p5 p3
  • 7v4 7v3 d. (kv3)v3
  • Solve each equation or inequality.
  • 121 111 n c. 343 74n 1
  • 33k 729 d. 5n2 625

Answers 1)a) 218 b) 7v4 v3 c) p8 d) k3 2)a) n
1 b) n 2 c) n 1 d) n 2
6
10-2 Logarithms and Logarithmic Functions
  • A logarithm is an equation in the from logbn p
    where b ? 1, b gt 0, n gt 0, and bp n.
  • Exponential Equation Logarithmic Equation
  • n bp p logbn
  • exponent or logarithm
  • base
  • number
  • Example x 63 can be re-written as 3 log6 x
  • ³/2 log2 x can be re-written as x 23/2

7
10-2 Logarithms and Logarithmic Functions
  • A logarithmic function has the from y logb,
    where b gt 0 and b ? 1.
  • The exponential function y bx and the
    logarithmic function y logb are inverses of
    each other. This means that their composites are
    the identity function, or they form an equation
    with the form y logb bx is equal to x.
  • Example log5 53 3
  • 2log2 (x 1) x 1

8
10-2 Logarithms and Logarithmic Functions
  • Property of Equality for Logarithmic Functions
    Given that b gt 0 and b ? 1, then logb x1 logb
    x2 if and only if x1 x2.
  • Example log8 (k2 6) log8 5k
  • k2 6 5k
  • k2 5k 6 0
  • (k 6)(k 1) 0
  • k 6 or k -1

9
10-2 Practice
  • Evaluate each expression.
  • log3 ½7 c. log5 625
  • log7 49 d. log4 64
  • Solve each equation.
  • log3 x 2 d. log12 (2p2) log12 (10p
    8)
  • log5 (t 4) log5 9t e. log2 (log4 16) x
  • logk 81 4 f. log9 (4r2) log9(36)

Answers 1)a) -3 b) 2 c) 4 d) 3 2)a) 9 b) ½ c) 3
d) 1, 5 e) 1 f) -3, 3
10
10-3 Properties of Logarithms
  • Product Property of Logarithms
    logb mn logb m logb n as long as m, n,
    and b are positive and b ? 1.
  • Example Given that log2 5 2.322, find log2 80
  • log2 80 log2 (24 5)
  • log2 24 log2 5 4 2.322 6.322
  • Quotient Property of Logarithms As long as m, n,
    and b are positive numbers and b ? 1, then logb
    m/n logb m logb n
  • Example Given that log3 6 1.6309, find log3
    6/81
  • log3 6/81 log3 6/34 log3 6 log3 34
  • 1.6309 4 -2.3691

11
10-3 Properties of Logarithms
  • Power Property of Logarithms For any real number
    p and positive numbers m and b, where b ? 1, logb
    mp plogb m
  • Example Solve ½ log4 16 2log4 8 log4 x
  • ½ log4 16 2log4 8 log4 x
  • log4 161/2 log4 82 log4 x
  • log4 4 log4 64 log4 x
  • log4 4/64 log4 x
  • x 4/64
  • x 1/16

12
10-3 Practice
  • Given log4 5 1.161 and log4 3 0.792, evaluate
    the following
  • log4 15 b. log4 192
  • log4 5/3 d. log4 144/25
  • Solve each equation.
  • 2 log3 x ¼ log2 256
  • 3 log6 2 ½ log6 25 log6 x
  • ½ log4 144 log4 x log4 4
  • 1/3 log5 27 2 log5 x 4 log5 3

Answers 1)a) 1.953 b) 3.792 c) 0.369 d) 1.544
2)a) x 2 b) x 8/5 c) x 36 d) x 3v3
13
10-4 Common Logarithms
  • Logarithms in base 10 are called common
    logarithms. They are usually written without the
    subscript 10.
  • Example log10 x log x
  • The decimal part of a log is the mantissa and the
    integer part of the log is called the
    characteristic.
  • Example log (3.4 x 103) log 3.4 log 103
  • 0.5315 3
  • mantissa characteristic

14
10-4 Common Logarithms
  • In a log we are given a number and asked to find
    the logarithm, for example log 4.3. When we are
    given the logarithm and asked to find the log, we
    are finding the antilogarithm.
  • Example log x 2.2643
  • x 10 2.2643
  • x 183.78
  • Example log x 0.7924
  • x 10 0.7924
  • x 6.2

15
10-4 Practice
  • If log 3600 3.5563, find each number.
  • mantissa of log 3600 d. log 3.6
  • characteristic of log 3600 e. 10 3.5563
  • antilog 3.5563 f. mantissa of log 0.036
  • Find the antilogarithm of each.
  • 2.498 c. -1.793
  • 0.164 d. 0.704 2

Answers 1)a) 0.5563 b) 3 c) 3600 d) 0.5563 e)
3600 f) 0.5563 2)a) 314.775 b) 1.459 c) 0.016 d)
0.051
16
10-5 Natural Logarithms
  • e is the base for the natural logarithms, which
    are abbreviated ln. Natural logarithms carry the
    same properties as logarithms.
  • e is an irrational number with an approximate
    value of 2.718. Also, ln e 1.

17
10-5 Practice
  • Find each value rounded to four decimal places.
  • ln 6.94 e. antiln -3.24
  • ln 0.632 f. antiln 0.493
  • ln 34.025 g. antiln -4.971
  • ln 0.017 h. antiln 0.835

Answers 1)a) 1.9373 b) -0.4589 c) 3.5271 d)
-4.0745 e) 0.0392 f) 1.6372 g) 0.0126 h) 2.3048
18
10-6 Solving Exponential Equations
  • Exponential equations are equations where the
    variable appears as an exponent. These equations
    are solved using the property of equality for
    logarithmic functions.
  • Example 5x 18
  • log 5x log 18
  • x log 5 log 18
  • x log 18
  • log 5
  • x 1.796

19
10-6 Solving Exponential Equations
  • When working in bases other than base 10, you
    must use the Change of Base Formula which says
    loga n logb n
  • logb a
  • For this formula a, b, and n are positive
    numbers where a ? 1 and b ? 1.
  • Example log7 196
  • log 196 change of base formula
  • log 7 a 7, n 196, b 10
  • 2.7124

20
10-6 Practice
  • Find the value of the logarithm to 3 decimal
    places.
  • log7 19 c. log3 91
  • log12 34 d. log5 48
  • Use logarithms to solve each equation. Round to
    three decimal places.
  • 13k 405 c. 5x-2 6x
  • 6.8b-3 17.1 d. 362p1 14p-5

Answers 1)a) 1.513 b) 1.419 c) 4.106 d) 2.405
2)a) k 2.341 b) B 4.481 c) x -17.655 d) p
-3.705
21
10-7 Growth and Decay
  • The general formula for growth and decay is y
    nekt, where y is the final amount, n is the
    initial amount, k is a constant, and t is the
    time.
  • To solve problems using this formula, you will
    apply the properties of logarithms.

22
10-7 Practice
  • Population Growth The town of Bloomington-Normal,
    Illinois, grew from a population of 129,180 in
    1990, to a population of 150,433 in 2000.
  • Use this information to write a growth equation
    for Bloomington-Normal, where t is the number of
    years after 1990.
  • Use your equation to predict the population of
    Bloomington-Normal in 2015.
  • Use your equation to find the amount year when
    the population of Bloomington-Normal reaches
    223,525.

23
10-7 Practice Solution
  • Use this information to write a growth equation
    for Bloomington-Normal, where t is the number of
    years after 1990.
  • y nekt
  • 150,433 (129,180)ek(10)
  • 1.16452 e10k
  • ln 1.16452 ln e10k
  • 0.152311 10k
  • k 0.015231
  • equation y 129,180e0.015231t

24
10-7 Practice Solution
  • Use your equation to predict the population of
    Bloomington-Normal in 2015.
  • y 129,180e0.015231t
  • y 129,180e(0.015231)(25)
  • y 129,180e0.380775
  • y 189,044

25
10-7 Practice Solution
  • Use your equation to find the amount year
  • when the population of Bloomington-Normal
  • reaches 223,525.
  • y 129,180e0.015231t
  • 223,525 129,180e0.015231t
  • 1.73034 e0.015231t
  • ln 1.73034 ln e0.015231t
  • 0.548318 0.015231t
  • t 36 years
  • 1990 36 2026
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