Worksheet 61 (11
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) [pic]
[pic]
[pic]
[pic] The solution set is { }.
b) [pic]
[pic]
[pic]
[pic]
[pic] The solution set is { }.
Worksheet 61 (11.1)
Problems - Solve:
1. [pic]
2. [pic]
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 ΦNote: This is an example of an increasing function.
| | |
|x |f(x) |
| | |
|-2 |1/9 |
| | |
|-1 | |
| | |
|0 | |
| | |
|1 | |
| | |
|2 | |
f(x)
[pic]x
Worksheet 61 (11.1)
b) f(x) =[pic] ΦNote: This is an example of a decreasing function.
| | |
|x |f(x) |
| | |
|-2 |9 |
| | |
|-1 | |
| | |
|0 | |
| | |
|1 | |
| | |
|2 | |
f(x)
[pic]x
c) f(x) = 3x + 2 ΦNote: This is a horizontal translation of f(x) = 3x.
| | |
|x |f(x) |
| | |
|-2 |1 |
| | |
|-1 | |
| | |
|0 | |
| | |
|1 | |
| | |
|2 | |
f(x)
[pic]x
Worksheet 61 (11.1)
d) f(x) =[pic]
ΦNote:This is a vertical translation of [pic].
| | |
|x |f(x) |
| | |
|-2 |11 |
| | |
|-1 | |
| | |
|0 | |
| | |
|1 | |
| | |
|2 | |
f(x)
[pic]x
Problems - Graph:
3. [pic] [pic]
4. [pic] [pic]
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:
[pic];
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.
[pic]
[pic]
[pic]
[pic]
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.
Worksheet 62 (11.2)
Summary 2:
As n gets infinitely large, the expression[pic]approaches 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.
[pic]
[pic]
[pic]
There were __________ bacteria initially.
Problem
2. Find the total accumulated money for $2000 invested at 12% compounded continuously for 5 years.
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.[pic] 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) [pic] is equivalent to[pic].
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[pic].
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 =[pic]
[pic]= x
[pic]= 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 =[pic]
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 + _______________
= __________
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[pic]= 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[pic].
log 10[pic] = log 10 101 - __________
= __________
b) If log 8 5 = 0.7740, evaluate log 8[pic]. (Recall: 82 = 64)
log 8[pic]= log 8 64 - __________
= ______ - .7740
= __________
Problems
7. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10[pic].
Worksheet 63 (11.3)
8. If log 2 5 = 2.3219, evaluate log 2[pic].
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[pic].
log 10[pic] = ( ) log 10 1995
= __________
b) Express as a simpler logarithmic expression: [pic]
[pic]=[pic]
=[pic]
=[pic]
c) Solve: log 3 (2x - 1) + log 3 (x + 1) = 2
log 3 ( )( ) = 2
32 = ( )( )
9 = _______________
0 = _______________
x = _____ or x = _____
ΦNote: Logarithms are only defined for positive numbers. Negative results are extraneous.
The solution set is { }.
Worksheet 63 (10.3)
Problems
9. If log 10 5 = 0.6990, evaluate log 10 54.
10. Express as a simpler logarithmic expression: [pic]
11. Solve: log 5 (4x + 1) - log 5 (x - 1) = 1
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
ΦNote: 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.
[pic]
b) Graph: f(x) = log 3 (x - 2)
ΦNote: This is a horizontal translation 2 units right.
[pic]
Worksheet 64 (11.4)
Problems
1. Graph: f(x) =[pic] ΦNote: See warm-up 2b in section 10.1.
[pic]
2. Graph: f(x) = 2 +[pic]
[pic]
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 = __________
Worksheet 64 (11.4)
Problems - Evaluate to four decimal places:
3. log 0.0243 4. log 0.243
5. log 2.43 6. log 24.3
Warm-up 3. Find x to five significant digits:
a) log x = 0.4150
ΦNote: 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 = 0.0101 8. 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 = __________
Worksheet 64 (11.4)
Problems - Evaluate to four decimal places:
9. ln 0.0243 10. ln 0.243
11. ln 2.43 12. ln 24.3
Warm-up 5. Find x to five significant digits:
a) ln x = 0.4150
ΦNote: 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 = 0.0101 14. ln x = -4.321
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 =[pic]
x = __________ The solution set is { }.
b) 5x + 1 = 7
log ( ) = log ( )
( )log 5 = log ( )
x + 1 =[pic]
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
Worksheet 65 (11.5)
3. log (x - 1) + log (x - 4) = 1
Warm-up 2. Use the compound interest formula A =[pic] 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 =[pic]
[pic]
[pic]
[pic]
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.)
Worksheet 65 (11.5)
Summary 2:
The change-of-base formula for logarithms:
log a r =[pic]; 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 =[pic]
≈ __________
ΦNote: 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 =[pic]
≈ __________
Problems - Approximate to 3 decimal places:
5. log 6 88 6. log 2 0.001
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