1. Exploring Exponential and Logarithmic Functions
Chapter 10
Exploring Exponential and Logarithmic Functions
By Kathryn Valle

2. 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.

3. 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

4. 10-1 Examples Solve: 128 = 24n – 1 53n + 2 > 625

5. 10-1 Practice Simplify each expression:
(23)6 c. p5 + p3
7√4 + 7√3 d. (k√3)√3
Solve each equation or inequality.
121 = n c. 343 = 74n – 1
33k = d. 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

6. 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 Equation Logarithmic 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

7. 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

8. 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

9. 10-2 Practice Evaluate each expression. Solve each equation.
log3 ½7 c. log5 625
log d. log4 64
Solve each equation.
log3 x = d. log12 (2p2) – log12 (10p – 8)
log5 (t + 4) = log5 9t e. log2 (log4 16) = x
logk 81 = 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. 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)
= log log2 5 ≈ ≈ 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 ≈ , find log3 6/81:
log3 6/81 = log3 6/34 = log3 6 – log3 34
≈ – 4 ≈

11. 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
log4 4/64 = log4 x
x = 4/64
x = 1/16

12. 10-3 Practice
Given log4 5 ≈ and log4 3 ≈ 0.792, evaluate the following:
log b. log4 192
log4 5/3 d. log4 144/25
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 log log5 x = 4 log5 3
Answers: 1)a) b) c) d) )a) x = ± 2 b) x = 8/5 c) x = 36 d) x = 3√3

13. 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 log 103 =
mantissa characteristic

14. 10-4 Common Logarithms
In a log we are given a number and asked to find the logarithm, for example log When we are given the logarithm and asked to find the log, we are finding the antilogarithm.
Example: log x =
x =
x =
Example: log x =
x =
x = 6.2

15. 10-4 Practice If log 3600 = 3.5563, find each number.
mantissa of log d. log 3.6
characteristic of log e
antilog f. mantissa of log 0.036
Find the antilogarithm of each. c
d – 2
Answers: 1)a) b) 3 c) 3600 d) e) 3600 f) )a) b) c) d) 0.051

16. 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 Also, ln e = 1.

17. 10-5 Practice Find each value rounded to four decimal places.
ln e. antiln -3.24
ln f. antiln 0.493
ln g. antiln
ln h. antiln 0.835
Answers: 1)a) b) c) d) e) f) g) h)

18. 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

19. 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 change of base formula
log 7 a = 7, n = 196, b = 10 ≈

20. 10-6 Practice Find the value of the logarithm to 3 decimal places.
log c. log3 91
log d. log5 48
Use logarithms to solve each equation. Round to three decimal places.
13k = c. 5x-2 = 6x
6.8b-3 = d. 362p+1 = 14p-5
b) B = c) x = d) p =
Answers: 1)a) b) c) d) )a) k = 2.341

21. 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.

22. 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.

23. 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)
= e10·k
ln = ln e10·k
= 10·k
k =
equation: y = 129,180·e ·t

24. 10-7 Practice Solution
Use your equation to predict the population of Bloomington-Normal in 2015.
y = 129,180·e ·t
y = 129,180·e( )(25)
y = 129,180·e
y = 189,044

25. 10-7 Practice Solution Use your equation to find the amount year
when the population of Bloomington-Normal
reaches 223,525.
y = 129,180·e ·t
223,525 = 129,180·e ·t
= e ·t
ln = ln e ·t
= ·t
t = 36 years
= 2026