Chapter 10
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Chapter 10. Exploring Exponential and Logarithmic Functions. By Kathryn Valle. 10-1 Real Exponents and Exponential Functions. An exponential function is any equation in the form y = a · b x where a ≠ 0, b > 0, and b ≠ 1. b is referred to as the base.

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Chapter 10

Chapter 10

Exploring Exponential and Logarithmic Functions

By Kathryn Valle


10 1 real exponents and exponential functions

10-1 Real Exponents and Exponential Functions

  • An exponential function is any equation in the form y = a·bx where a ≠ 0, b > 0, and b ≠ 1. b is referred to as the base.

  • Property of Equality for Exponential Functions: If in the equation y = a·bx, b is a positive number other than 1, then bx1 = bx2 if and only if x1 = x2.


10 1 real exponents and exponential functions1

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

      5√2 · 5√7 = 5√2 + √7

  • Power of a Power Property: To simplify a term with an exponent and raised to another power, multiply the exponents.

    • Example: (43)2 = 46

      (8√5)4 = 84·√5


10 1 examples

10-1 Examples

  • Solve:

    128 = 24n – 1 53n + 2> 625

    27 = 24n –153n + 2> 54

    7 = 4n – 13n + 2 > 4

    8 = 4n 3n > 2

    n = 2 n > ²/³


10 1 practice

10-1 Practice

  • Simplify each expression:

    • (23)6c. p5 + p3

    • 7√4 + 7√3d. (k√3)√3

  • Solve each equation or inequality.

    • 121 = 111 + nc. 343 = 74n – 1

    • 33k = 729d. 5n2 = 625

Answers: 1)a) 218 b) 7√4 + √3 c) p8 d) k3 2)a) n = 1 b) n = 2 c) n = 1 d) n = ±2


10 2 logarithms and logarithmic functions

10-2 Logarithms and Logarithmic Functions

  • A logarithm is an equation in the from logbn = p where b ≠ 1, b > 0, n > 0, and bp = n.

  • Exponential EquationLogarithmic 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


10 2 logarithms and logarithmic functions1

10-2 Logarithms and Logarithmic Functions

  • A logarithmic function has the from y = logb, where b > 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


10 2 logarithms and logarithmic functions2

10-2 Logarithms and Logarithmic Functions

  • Property of Equality for Logarithmic Functions:Given that b > 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


10 2 practice

10-2 Practice

  • Evaluate each expression.

    • log3½7c. 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 3 properties of logarithms

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)

      = log224 + 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 log36/81:

      log36/81 = log36/34= log3 6 – log3 34

      ≈ 1.6309 – 4 ≈ -2.3691


10 3 properties of logarithms1

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 = p·logb m

    • Example: Solve ½ log4 16– 2·log4 8 = log4 x

      ½ log4 16– 2·log4 8 = log4 x

      log4 161/2 – log4 82 = log4 x

      log4 4 – log4 64 = log4 x

      log44/64 = log4 x

      x = 4/64

      x = 1/16


10 3 practice

10-3 Practice

  • Given log4 5 ≈ 1.161 and log4 3 ≈ 0.792, evaluate the following:

    • log4 15b. log4 192

    • log4 5/3d. log4144/25

  • 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 = 3√3


10 4 common logarithms

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


10 4 common logarithms1

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


10 4 practice

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.

    • 2.498c. -1.793

    • 0.164d. 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


10 5 natural logarithms

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.


10 5 practice

10-5 Practice

  • Find each value rounded to four decimal places.

    • ln 6.94e. antiln -3.24

    • ln 0.632f. antiln 0.493

    • ln 34.025g. antiln -4.971

    • ln 0.017h. 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


10 6 solving exponential equations

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


10 6 solving exponential equations1

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 196change of base formula

      log 7a = 7, n = 196, b = 10

      ≈ 2.7124


10 6 practice

10-6 Practice

  • Find the value of the logarithm to 3 decimal places.

    • log7 19c. log3 91

    • log12 34d. log5 48

  • Use logarithms to solve each equation. Round to three decimal places.

    • 13k = 405c. 5x-2 = 6x

    • 6.8b-3 = 17.1d. 362p+1 = 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


10 7 growth and decay

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.


10 7 practice

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.


10 7 practice solution

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 = e10·k

    ln 1.16452 = ln e10·k

    0.152311 = 10·k

    k = 0.015231

    equation: y = 129,180·e0.015231·t


10 7 practice solution1

10-7 Practice Solution

  • Use your equation to predict the population of Bloomington-Normal in 2015.

    y = 129,180·e0.015231·t

    y = 129,180·e(0.015231)(25)

    y = 129,180·e0.380775

    y = 189,044


10 7 practice solution2

10-7 Practice Solution

  • Use your equation to find the amount year

    when the population of Bloomington-Normal

    reaches 223,525.

    y = 129,180·e0.015231·t

    223,525 = 129,180·e0.015231·t

    1.73034 = e0.015231·t

    ln 1.73034 = ln e0.015231·t

    0.548318 = 0.015231·t

    t = 36 years

    1990 + 36 = 2026


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