Exploring Exponential and Logarithmic Functions

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Chapter 10

• Exploring Exponential and Logarithmic Functions

By Kathryn Valle
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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.

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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

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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

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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
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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

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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

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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

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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
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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

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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

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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
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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

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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

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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
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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.

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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
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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

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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

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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
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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.

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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.

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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

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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

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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