Chapter 11 Exponential and Logarithmic Functions

Worksheet 61 (11.1)

Chapter 11 Exponential and Logarithmic Functions

11.1 Exponents and Exponential Functions

Summary 1:

The rules of exponents from chapter 5 can be extended to any real number exponent.

If b > 0, b  1, and m and n are real numbers, then bn = bm if and only if

n = m.

Warm-up 1. Solve:

a)

The solution set is { }.

b)

The solution set is { }.

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Worksheet 61 (11.1)

Problems - Solve:

1.

2.

Summary 2:

If b > 0 and b  1, then the function f defined by f(x) = bx, where x is any real number, is called the exponential function with base b.

Warm-up 2.Graph:

a) f(x) = 3x Note: This is an example of an increasing function.

x / f(x)
-2 / 1/9
-1
0
1
2

f(x)

x

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Worksheet 61 (11.1)

b) f(x) =Note: This is an example of a decreasing function.

x / f(x)
-2 / 9
-1
0
1
2

f(x)

x

c) f(x) = 3x + 2 Note: This is a horizontal translation of f(x) = 3x.

x / f(x)
-2 / 1
-1
0
1
2

f(x)

x

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Worksheet 61 (11.1)

d) f(x) =

Note:This is a vertical translation of .

x / f(x)
-2 / 11
-1
0
1
2

f(x)

x

Problems - Graph:

3.

4.

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Worksheet 62 (11.2)

11.2 Applications of Exponential Functions

Summary 1:

Compound interest is an example of an exponential function.

General formula for compound interest:

;

where P = principal, n = number of times being compounded,

t = number of years, r = rate of percent,

A = total amount of money accumulated.

Warm-up 1. a) Find the total amount of money accumulated for $2000 invested at 12% compounded quarterly for 5 years.

The total amount of accumulation is ______.

Problem

1. Find the total amount of money accumulated for $1500 invested at 10% compounded monthly for 3 years.

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Worksheet 62 (11.2)

Summary 2:

As n gets infinitely large, the expressionapproaches the number e, where e equals 2.71828 to five decimal places.

The function defined by f(x) = ex is the natural exponential function.

Note: Use the ex key on the calculator to find functional values for x.

Formulas involving e:

1. A = P ert Used for compounding continuously.

A = total accumulated value, P = principal, t = years, r = rate

2. Q(t) = Q0 ekt Used for growth-and-decay applications.

Q(t) = quantity of substance at any time,

Q0 = initial quantity of substance, k = constant, t = time

Warm-up 2. a) The number of bacteria present in a certain culture after t hours is given by the equation Q = Q0 e0.3t, where Q0 represents the number of bacteria initially. If 18,149 bacteria are present after 6 hours, find how many bacteria were present in the culture initially.

There were ______bacteria initially.

Problem

2. Find the total accumulated money for $2000 invested at 12% compounded continuously for 5 years.

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Worksheet 63 (11.3)

11.3 Logarithms

Summary 1:

If r is any positive real number, then the unique exponent t such that t = r is called the logarithm of r with base b and is denoted by log b r.

log b r = t is equivalent to bt = r.

For b > 0 and b  1, and r > 0,

1. log b b = 1 since b1 = b.

2. log b 1 = 0 since b0 = 1.

3. since log b r = t.

Warm-up 1. Find the equivalent exponential expression:

a) log 5 125 = 3 is equivalent to 5 ( ) = 125.

b) log 10 10000 = 4 is equivalent to 10 ( ) = 10000.

c) log 2 32 = 5 is equivalent to 2 ( ) = 32.

Warm-up 2. Find the equivalent logarithmic expression:

a) 10-3 = 0.001 is equivalent to log 10 0.001 = _____.

b) is equivalent to.

c) 54 = 625 is equivalent to log 5 625 = _____.

Problems

1. Find the exponential expressions for log 3 27 = 3 and log 10 .00001 = -5.

2. Find the logarithmic expressions for 102 = 100 and.

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Worksheet 63 (11.3)

Warm-up 3. a) Evaluate log 3 243 by first rewriting in exponential form and then solving. (See summary 1 in section 10.1.)

Let log 3 243 = x

This is equivalent to:3x = 243

3x = 3 ( )

x = _____

Therefore, log 3 243 = _____.

b) Solve: log 32 x =

= x

= x

_____ = x The solution set is { }.

Problems

3. Evaluate log 10 10000 by first rewriting in exponential form and then solving.

4. Solve:log 125 x =

Summary 2:

For positive real numbers b, r, and s where b  1,

log b rs = log b r + log b s

Warm-up 4. a) If log 10 2 = 0.3010 and log 10 7 = 0.8451, evaluate log 10 14.

log 10 14 = log 10 (2  _____ )

= log 10 2 + ______

= ______

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Worksheet 63 (11.3)

b) If log 2 7 = 2.8074 and log 2 5 = 2.3222, evaluate log 2 35.

log 2 35 = log 2 ( _____  _____ )

= ______+ ______

= ______

Problems

5. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10 15.

6. If log 2 7 = 2.8074 and log 2 3 = 1.5850, evaluate log 2 21.

Summary 3:

For positive real numbers b, r, and s where b  1,

log b= log b r - log b s

Warm-up 5.a) If log 10 101 = 2.0043 and log 10 23 = 1.3617, evaluate

log 10.

log 10 = log 10 101 - ______

= ______

b) If log 8 5 = 0.7740, evaluate log 8. (Recall: 82 = 64)

log 8= log 8 64 - ______

= ______- .7740

= ______

Problems

7. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10.

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Worksheet 63 (11.3)

8. If log 2 5 = 2.3219, evaluate log 2.

Summary 4:

For positive real numbers b, r, and p where b  1,

log b r p = p (log b r)

Warm-up 6. a) If log 10 1995 = 3.2999, evaluate log 10.

log 10 = ( ) log 10 1995

= ______

b) Express as a simpler logarithmic expression:

=

=

=

c) Solve: log 3 (2x - 1) + log 3 (x + 1) = 2

log 3 ( )( ) = 2

32 = ( )( )

9 = ______

0 = ______

x = _____ or x = _____

Note: Logarithms are only defined for positive numbers. Negative results are extraneous.

The solution set is { }.

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Worksheet 63 (10.3)

Problems

9. If log 10 5 = 0.6990, evaluate log 10 54.

10. Express as a simpler logarithmic expression:

11. Solve: log 5 (4x + 1) - log 5 (x - 1) = 1

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Worksheet 64 (11.4)

11.4 Logarithmic Functions

Summary 1:

A function defined by an equation of the form f(x) = log b x, b > 0 and b  1 is called a logarithmic function.

y = log b x is equivalent to x = by.

f(x) = bx and g(x) = log b x are inverse functions.

Warm-up 1. a) Graph: y = log 3 x

Note: This is the inverse of y = 3x from warm-up 2a in section 10.1. Inverses are reflections of each other through the line y = x.

b) Graph: f(x) = log 3 (x - 2)

Note: This is a horizontal translation 2 units right.

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Worksheet 64 (11.4)

Problems

1. Graph: f(x) = Note: See warm-up 2b in section 10.1.

2. Graph: f(x) = 2 +

Summary 2:

Logarithms with a base of 10 are called common logarithms.

log 10 x = log x

Note: Use log key on calculator to evaluate common logarithms.

f(x) = log x and g(x) = 10x are inverse functions.

Warm-up 2. Evaluate to four decimal places:

a) log 1.25 = ______

b) log 12.5 = ______

c) log 125 = ______

d) log 1250 = ______

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Worksheet 64 (11.4)

Problems - Evaluate to four decimal places:

3. log 0.0243 4. log 0.243

5. log 2.436. log 24.3

Warm-up 3. Find x to five significant digits:

a) log x = 0.4150

Note: Use 10x key on calculator to find x.

x = 10( )

x = ______

b) log x = 1.6135

x = 10( )

x = ______

Problems - Find x to five significant digits:

7. log x = 0.01018. log x = -4.321

Summary 3:

Natural logarithms are logarithms that have a base of e.

log e x = ln x

Note: Use ln key on calculator to evaluate natural logarithms.

f(x) = ln x and g(x) = ex are inverse functions.

Warm-up 4. Evaluate to four decimal places:

a) ln 1.25 = ______

b) ln 12.5 = ______

c) ln 125 = ______

d) ln 1250 = ______

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Worksheet 64 (11.4)

Problems - Evaluate to four decimal places:

9. ln 0.024310. ln 0.243

11. ln 2.4312. ln 24.3

Warm-up 5. Find x to five significant digits:

a) ln x = 0.4150

Note: Use ex key on calculator to find x.

x = e( )

x = ______

b) ln x = 1.6135

x = e( )

x = ______

Problems - Find x to five significant digits:

13. ln x = 0.010114. ln x = -4.321

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Worksheet 65 (10.5)

11.5 Exponential Equations, Logarithmic Equations, and

Problem Solving

Summary 1:

If x > 0, y > 0, and b  1, then x = y if and only if log b x = log b y.

Warm-up 1. Solve to the nearest hundredth:

a) 10x = 5

log ( ) = log ( )

( )log 10 = log 5

x =

x = ______The solution set is { }.

b) 5x + 1 = 7

log ( ) = log ( )

( )log 5 = log ( )

x + 1 =

x = ______The solution set is { }.

c) ln (x + 2) = ln (x - 3) + ln 2

ln (x + 2) = ln [2( )]

x + 2 = ______

x = _____ The solution set is { }.

Problems - Solve to the nearest hundredth:

1. ex + 1 = 40

2. 72x = 11

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Worksheet 65 (11.5)

3. log (x - 1) + log (x - 4) = 1

Warm-up 2. Use the compound interest formula A = and

logarithms to solve:

a) How long will it take $1000 to double itself if invested at 10% interest compounded quarterly? (Round to tenths.)

A =

t = ______ It will take _____ years.

Problem - Use the formula A = Pert and natural logarithms to solve:

4. How long will it take $1000 to double itself at 10% interest when compounded continuously? (Round to nearest tenth.)

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Worksheet 65 (11.5)

Summary 2:

The change-of-base formula for logarithms:

log a r =; where a, b, and r are positive real numbers

and a  1 and b  1.

Warm-up 3. Approximate to 3 decimal places:

a) log 3 15

log 3 15 =

 ______

Note: Either common or natural logarithms can be used to approximate logarithms with bases other than 10 or e.

b) log 5 0.004

log 5 0.004 =

 ______

Problems - Approximate to 3 decimal places:

5. log 6 886. log 2 0.001

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