- 70 Views
- Uploaded on
- Presentation posted in: General

Chapter 10

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Chapter 10

Exploring Exponential and Logarithmic Functions

By Kathryn Valle

- An exponential function is any equation in the form y = aÂ·bx where a â‰ 0, b > 0, and b â‰ 1. b is referred to as the base.
- Property of Equality for Exponential Functions: If in the equation y = aÂ·bx, b is a positive number other than 1, then bx1 = bx2 if and only if x1 = x2.

- Product of Powers Property: To simplify two like terms each with exponents and multiplied together, add the exponents.
- Example: 34 Â· 35 = 39
5âˆš2 Â· 5âˆš7 = 5âˆš2 + âˆš7

- Example: 34 Â· 35 = 39
- Power of a Power Property: To simplify a term with an exponent and raised to another power, multiply the exponents.
- Example: (43)2 = 46
(8âˆš5)4 = 84Â·âˆš5

- Example: (43)2 = 46

- Solve:
128 = 24n â€“ 1 53n + 2> 625

27 = 24n â€“153n + 2> 54

7 = 4n â€“ 13n + 2 > 4

8 = 4n 3n > 2

n = 2 n > Â²/Â³

- Simplify each expression:
- (23)6c. p5 + p3
- 7âˆš4 + 7âˆš3d. (kâˆš3)âˆš3

- Solve each equation or inequality.
- 121 = 111 + nc. 343 = 74n â€“ 1
- 33k = 729d. 5n2 = 625

Answers: 1)a) 218 b) 7âˆš4 + âˆš3 c) p8 d) k3 2)a) n = 1 b) n = 2 c) n = 1 d) n = Â±2

- A logarithm is an equation in the from logbn = p where b â‰ 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
Â³/2 = log2 x can be re-written as x = 23/2

- Example: x = 63 can be re-written as 3 = log6 x

- A logarithmic function has the from y = logb, where b > 0 and b â‰ 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 â€“ 1) = x â€“ 1

- Example: log5 53 = 3

- Property of Equality for Logarithmic Functions:Given that b > 0 and b â‰ 1, then logb x1 = logb x2 if and only if x1 = x2.
- Example: log8 (k2 + 6) = log8 5k
k2 + 6 = 5k

k2 â€“ 5k + 6 = 0

(k â€“ 6)(k + 1) = 0

k = 6 or k = -1

- Example: log8 (k2 + 6) = log8 5k

- Evaluate each expression.
- log3Â½7c. log5 625
- log7 49 d. log4 64

- Solve each equation.
- log3 x = 2 d. log12 (2p2) â€“ log12(10p â€“ 8)
- log5 (t + 4) = log5 9t e. log2 (log4 16) = x
- logk 81 = 4 f. log9 (4r2) â€“ log9(36)

Answers: 1)a) -3 b) 2 c) 4 d) 3 2)a) 9 b) Â½ c) 3 d) 1, 5 e) 1 f) -3, 3

- Product Property of Logarithms: logb mn = logb m + logb n as long as m, n, and b are positive and b â‰ 1.
- Example: Given that log2 5 â‰ˆ 2.322, find log2 80:
log2 80 = log2 (24Â· 5)

= log224 + log2 5 â‰ˆ 4 + 2.322 â‰ˆ 6.322

- Example: Given that log2 5 â‰ˆ 2.322, find log2 80:
- Quotient Property of Logarithms: As long as m, n, and b are positive numbers and b â‰ 1, then logb m/n = logb m â€“ logb n
- Example: Given that log3 6 â‰ˆ 1.6309, find log36/81:
log36/81 = log36/34= log3 6 â€“ log3 34

â‰ˆ 1.6309 â€“ 4 â‰ˆ -2.3691

- Example: Given that log3 6 â‰ˆ 1.6309, find log36/81:

- Power Property of Logarithms: For any real number p and positive numbers m and b, where b â‰ 1, logb mp = pÂ·logb m
- Example: Solve Â½ log4 16â€“ 2Â·log4 8 = log4 x
Â½ log4 16â€“ 2Â·log4 8 = log4 x

log4 161/2 â€“ log4 82 = log4 x

log4 4 â€“ log4 64 = log4 x

log44/64 = log4 x

x = 4/64

x = 1/16

- Example: Solve Â½ log4 16â€“ 2Â·log4 8 = log4 x

- Given log4 5 â‰ˆ 1.161 and log4 3 â‰ˆ 0.792, evaluate the following:
- log4 15b. log4 192
- log4 5/3d. log4144/25

- Solve each equation.
- 2 log3 x = Â¼ log2 256
- 3 log6 2â€“ Â½ log6 25 = log6 x
- Â½ log4 144â€“ 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 = Â± 2 b) x = 8/5 c) x = 36 d) x = 3âˆš3

- 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

- Example: log (3.4 x 103) = log 3.4 + log 103

- 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

- Example: log x = 2.2643

- 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 â€“ 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

- 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.

- 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

- 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 Â· log 5 = log 18

x = log 18

log 5

x = 1.796

- Example: 5x = 18

- 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 â‰ 1 and b â‰ 1.

- Example: log7 196
log 196change of base formula

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

â‰ˆ 2.7124

- Example: log7 196

- 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

- 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.

- 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.

- 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)Â·ek(10)

1.16452 = e10Â·k

ln 1.16452 = ln e10Â·k

0.152311 = 10Â·k

k = 0.015231

equation: y = 129,180Â·e0.015231Â·t

- Use your equation to predict the population of Bloomington-Normal in 2015.
y = 129,180Â·e0.015231Â·t

y = 129,180Â·e(0.015231)(25)

y = 129,180Â·e0.380775

y = 189,044

- Use your equation to find the amount year
when the population of Bloomington-Normal

reaches 223,525.

y = 129,180Â·e0.015231Â·t

223,525 = 129,180Â·e0.015231Â·t

1.73034 = e0.015231Â·t

ln 1.73034 = ln e0.015231Â·t

0.548318 = 0.015231Â·t

t = 36 years

1990 + 36 = 2026