DATE
NAME
10-1
Student Edition
Pages 596–602
Study Guide
Real Exponents and Exponential Functions
Refer to the following definitions and example when working
with real exponents. All the properties of rational exponents
that you know apply to real exponents as well. Recall that
am ? an 5 am 1 n, (am)n 5 amn, and am 4 an 5 am 2 n (when a Þ 0).
Definition of Exponential Function
An equation of the form y 5 a ? b x, where a Þ 0, b . 0,
and b Þ 1, is called an exponential function with base b.
Property of Equality for Exponential Functions
Definition
Example
Suppose b is a positive number
other than 1. Then
b x1 5 b x2 if and only if x1 5 x2.
Solve 2
25
5
6
2
5
5
5
5
5
5
3n 2 1
2
for n.
23n 2 1
3n 2 1
Property of Equality
3n
for Exponential Functions
n
Simplify each expression.
–
–
–
–
–
–
–
—
2. 8Ï5 4 2Ï5 4Ï5
1. (3Ï2)Ï2 9
–
–
–
–
5. 25Ï2 ? 125Ï2
4. (xÏ2 y3Ï2)Ï2
–
–
55Ï2 or 3125Ï2
x2 y6
–
3. (mÏ28)Ï7 m14
—
6. (cÏ6)Ï63
—
c3Ï42
Solve each equation.
7. 2x 1 1 5 22x 1 3 22
10. 22x 1 1 5 42x 1 2 2
© Glencoe/McGraw-Hill
3
2
8. 4x 1 1 5 82x 1 3 2
11. 6y 5 63y 1 1 2
T69
1
2
7
4
1
1
9. 32x 2 1 5 9 2
2
12. 82n 5 16n 2 3 26
Algebra 2
DATE
NAME
10-2
Student Edition
Pages 605–610
Study Guide
Logarithms and Logarithmic Functions
Logarithmic functions are the inverses of exponential functions.
Exponential Equation
Logarithmic Equation
n5b
p 5 logb n
p
Definition of Logarithms
Definition
Example
Suppose b . 0 and b Þ 1. For n . 0,
there is a number p such that
logbn 5 p if and only if b p 5 n.
Solve log2 x 5 3 for x.
log2 x 5 3
23 5 x
Definition of Logarithms
x58
You can use the Property of Logarithmic Functions to solve
exponential functions involving logarithms.
Property of Equality for Logarithmic Functions
Definition
Suppose b . 0 and b Þ 1.
Then logb x1 5 logb x2 if and only
if x1 5 x2.
Example
Solve log3(2x 2 4) 5 log3(7x 1 1).
log3(2x 2 4) 5 log3(7x 1 1)
2x 2 4 5 7x 1 1 Property of Equality for
25x 5 5
Logarithmic Functions
x 5 21
Evaluate each expression.
1. log4 64 3
2. log2 64 6
3. log10 100,000 5
4. log5 625 4
5. log3 27 3
6. log11 121 2
Solve each equation.
7. log5 m 5 4 625
8. log2 32 5 3x
10. log4(3x 2 1) 5 log4(2x 1 3)
9. log3 2c 5 22
1
18
11. log2(x2 2 6) 5 log2(2x 1 2)
4, 22
4
© Glencoe/McGraw-Hill
5
3
T70
Algebra 2
DATE
NAME
10-3
Student Edition
Pages 611–616
Study Guide
Properties of Logarithms
Logarithms are exponents. Thus, the properties of logarithms
can be derived from the properties of exponents.
Properties of Logarithms
For all positive numbers m, n, and b, where b Þ 1.
Property
Example
Product Property
logb mn 5 logb m 1 logb n
Quotient Property
logb
m
5 logb
n
m 2 logb n
Given log3 5 5 1.465, find log3 25.
log3 25 5 log3(5 ? 5)
5 log3 5 1 log3 5
5 1.465 1 1.465 or 2.930
Given log6 2 5 0.3869 and log6 20 5 1.6720,
find log6 0.1.
log6 0.1 5 log6
2
( 20 )
5 log6 2 2 log6 20
5 0.3869 2 1.6720 or 21.2851
Power Property
logb m p 5 p ? logb m
Solve 2 log4 x 2 log4 8 5 log4 8.
2 log4 x 2 log4 8
2 log4 x
2 log4 x
2 log4 x
log4 x 2
x2
x
5
5
5
5
5
5
5
log4
log4
log4
log4
log4
64
68
8
8 1 log4 8
(8 ? 8)
64
64
Use log12 3 5 0.4421 and log12 7 5 0.7831 to evaluate each
expression.
1. log12 21
1.2252
7
2. log12 3
3. log12 49
1.5662
0.3410
4. log12 36
5. log12 48
1.4421
1.5579
36
6. log12 49
0.1241
Solve each equation.
7. log2 4 2 log2(x 1 3) 5 log2 8
2
8. log10(x 1 3) 2 log10(2x 2 1) 5 log10 2
5
2
5
3
9. log3(c 1 3) 2 log3(4c 2 1) 5 log3 5
10. log2 x 2 3 log2 5 5 2 log2 10
8
19
© Glencoe/McGraw-Hill
12,500
T71
Algebra 2
10-4
DATE
NAME
Student Edition
Pages 617–621
Study Guide
Common Logarithms
Logarithms to base 10 are called common logarithms. Every
logarithm has two parts, the mantissa and the characteristic.
Logarithms depend on the use of scientific notation. The mantissa
is the logarithm of a number between 1 and 10. The characteristic
is the power of ten that is used when the number is expressed in
scientific notation.
A scientific calculator can be used to find
Example: Find log 273.
common logarithms. Use the LOG key.
ENTER: 273 LOG 2.4361626
The value of log 273 is approximately 2.4362.
In some cases, the calculator will display a
negative number. To avoid a negative mantissa,
you can add and subtract 10, in effect adding 0.
Sometimes a logarithm is given and you must
find the number. The number is called the
antilogarithm.
Example: Find log 0.0034.
ENTER: 0.0034 LOG 22.4685211
The value of log 0.0034 is approximately 22.4685.
22.4685 1 10 2 10 5 7.5315 2 10
Example: If log x 5 3.6355, find x.
ENTER: 3.6355 INV
LOG 4320.161676
The value of x is approximately 4320.
If log x 5 a, then x 5 antilog a.
You can use the INV and LOG keys
to find the antilog of a logarithm.
Use a scientific calculator to find the logarithm for each
number rounded to four decimal places. Then state the
characteristic and the mantissa.
1. 286.1
2.4565; 2; 0.4565
4. 0.496
20.3045; 21; 0.6955
2. 0.0048
3. 72.68
22.3188; 23; 0.6812
1.8613; 1; 0.8613
5. 6.15
6. 0.0000008
26.0969; 27; 0.9031
0.7889; 0; 0.7889
Use a scientific calculator to find the antilogarithm of each
logarithm rounded to four decimal places.
7. 2.162
145.2112
10. 22.353
0.0044
© Glencoe/McGraw-Hill
8. 21.42
9. 3.493
11. 0.681
12. 4.111
0.0380
4.7973
T72
3111.7163
12,912.1927
Algebra 2
DATE
NAME
10-5
Student Edition
Pages 622–625
Study Guide
Natural Logarithms
The number e is used extensively in science and mathematics. It is
an irrational number whose value is approximately 2.718. e is the
base for the natural logarithms. All the properties of common
logarithms apply also to natural logarithms. You can use a scientific
calculator to compute with natural logarithms. The key marked LN
is the natural logarithm key.
Example: Find ln 2.856 using a calculator.
ENTER: 2.856 LN 1.0494220
The natural logarithm of 2.856 is approximately 1.0494.
You can also find the antilogarithms of natural logarithms.
Example: Find x if ln x 5 2.874.
ENTER: 2.874 2nd e x 748.169500
So x is about 748.
Use a scientific calculator to find each value, rounded to four
decimal places.
1. ln 732
6.5958
4. ln 0.735
20.3079
7. ln 2.388
0.8705
10. antiln 1.3475
3.8478
13. antiln 0.0813
2. ln 1685
3. ln 84,350
7.4295
11.3427
5. ln 100
6. ln 0.0824
4.6051
8. ln 128,245
9. ln 0.00614
11.7617
25.0929
11. antiln 2.3862
12. antiln 0.5384
10.8721
14. antiln 4.3165
1.0847
74.9259
16. antiln 3.111
17. antiln 0.113
22.4435
1.1196
© Glencoe/McGraw-Hill
22.4962
T73
1.7133
15. antiln 2.4
11.0232
18. antiln 10
22026.4658
Algebra 2
10-6
DATE
NAME
Student Edition
Pages 626–630
Study Guide
Solving Exponential Equations
An equation with a variable in an exponent is
called an exponential equation. Such an
equation can be solved by using the Property of
Equality for Logarithmic Functions.
Example: Solve 3x 5 16.
3x 5 16
log 3x 5 log 16
x log 3 5 log 16
x5
log 16
log 3
x5
1.2041
0.4771
x 5 2.5238
The Change of Base Formula can be used to
find logarithms with bases other than 10. The
computations can be done using a calculator.
Example: Express log3 35 in terms
of common logarithms.
Then find its value.
Change of Base Formula
For all positive numbers a, b, and n, where a Þ 1
and b Þ 1, then loga n 5
logb n
.
logb a
log3 35 5
log 35
log 3
5
1.5441
0.4771
5 3.2364
Approximate the value of each logarithm to three decimal
places.
1. log2 4
2.000
4. log12 2
0.279
2. log5 75
3. log1.3 67
2.683
16.026
5. log14 126
1.833
6. log7 896
3.493
Use logarithms to solve each equation.
7. 4x 5 80
8. 5y 5 10
10. 1.32x 5 78
11. 63x 1 1 5 8
3.1611
8.3028
© Glencoe/McGraw-Hill
1.4306
0.0535
T74
9. 5x 5 18.5
1.8127
12. 2x 5 5x 2 2
3.5132
Algebra 2
DATE
NAME
10-7
Student Edition
Pages 631–636
Study Guide
Growth and Decay
Many problems can be solved by applying the following formulas:
Growth and Decay
Formula
y 5 ne kt
y is the final amount, n is the initial amount,
k is a constant, and t represents time.
Continuously
Compounded Interest
A 5 Pe r t
P is the initial investment, r is the annual
interest rate, and t is the time in years.
Value of Equipment and
Assets in Business
Vn 5 P (1 1 r )n
Vn is the new value, P is the initial value,
r is the fixed rate of appreciation or
depreciation, and n is the number of years.
Example: Find how long it will take money to double if it is invested
at 8% annual interest, compounded continuously.
A
2
ln 2
ln 2
ln 2
5
5
5
5
5
ln 2
0.08
5t
Pert
1e0.08t
ln e0.08 t
0.08t(ln e)
0.08t
Substitute 2 for A, 1 for P since the
amount is doubled. r 5 0.08
Take the natural log of each side.
Power Property of Logarithms
8.6643 5 t
The money will double in approximately 8.66 years.
Solve.
1. Carl plans to invest $500 at 8.25% interest, compounded
continously. How long will it take for his money to triple?
13.316 years
2. A certain strain of bacteria grows from 40 to 326 in 120
minutes. Find k for the growth formula.
0.0175
3. A $40,000 car depreciates at a constant rate of 12% per year.
In how many years will the car be worth $12,000?
9.42 years
© Glencoe/McGraw-Hill
T75
Algebra 2