1. College Algebra: Lesson 3
Exponential and Logarithmic Functions

2. Topics Topics include exponential functions logarithmic functions
properties of logarithms
exponential and logarithmic equations
modeling with exponential and logarithmic functions

3. Exponential Functions
Defined by f(x) = bx where
b > 0 and
b ≠ 1
Domain consists of all real numbers.
Range consists of all positive numbers.
One-to-one function.
Inverse is a function.

4. Exponential Functions
Graphs pass through (0, 1).
Y-intercept is 1.
There is no x-intercept.
Graph approaches, but does not touch, the x-axis.
If b > 1, graph goes up to the right. The greater the value of b, the steeper the increase of the graph.
If 0 < b < 1, graph goes down to the right. The smaller the value of b, the steeper the decrease.

5. The Natural Base e e is e is called the natural base.
an irrational number
defined as the value that (1 + 1/n)n approaches as n →∞ ≈
e is called the natural base.
The function f(x) = ex is called the natural exponential function.

6. Graphs of Exponential Functions
b > 1
0 < b < 1
Graphs of Exponential Functions

7. Formulas for Compound Interest
n compoundings per year
Continuous compounding
A = P(1 + r/n)nt
A = Pert
Formulas for Compound Interest

8. Logarithmic Functions
For x > 0 and b > 0, b ≠ 1,
y = logb x is equivalent to by = x.
The domain of f(x) = logbx consists of all positive real numbers.
The range of f(x) = logbx consists of all real numbers.

9. Logarithmic Functions
The graphs of all logarithmic functions pass through (1, 0).
The x-intercept is 1.
There is no y-intercept.
The graph of f(x) = logb x approaches, but does not touch, the y-axis.
If b > 1, the graph of f(x) = logb x goes up to the right.
If 0 < b < 1, the graph of f(x) = logb x goes down to the right.

10. Special Logarithms The logarithmic function with base 10.
Common Logarithms
Natural Logarithms
The logarithmic function with base 10.
f(x) = log10 x is usually written f(x) = log x
The logarithmic function with base e is
expressed as ln x.
read “el en of x.”
Special Logarithms

11. Graphs of Logarithmic Functions
b > 1
0 < b < 1
Graphs of Logarithmic Functions

12. Graphs of Exponential and Logarithmic Functions
The graphs of an exponential function (green) and its inverse, a logarithmic function (red), are reflections about y = x.

13. Examples Write log5125 = 3 in exponential form. 53=125
Log to exponential
Exponential to log
Write log5125 = 3 in exponential form.
53=125
Write √36 = 6 in logarithmic form.
√36 = 6 can be rewritten as 361/2 = 6. In logarithmic form, this would be log366= 1/2
Examples

14. Basic Logarithmic Properties Involving 1
logb b = 1
since b1 = b
logb 1 = 0
since b0 = 1

15. Inverse Properties of Logarithms
Exponential
The logarithm with base b of b raised to a power equals that power:
logb bx = x
b raised to the logarithm with base b of a number equals that number:
blogbx = x
Inverse Properties of Logarithms

16. Properties of Natural Logarithms
General Logarithms
Natural Logarithms
logb 1 = 0
logb b = 1
logb bx = x
blogbx = x
ln 1 = 0
ln e = 1
ln ex = x
eln x = x
Properties of Natural Logarithms

17. More Properties of Logarithms
Product Rule
Quotient Rule
The logarithm of a product is the sum of the logarithms.
The logarithm of a quotient is the difference of the logarithms.
More Properties of Logarithms

18. Still More Properties of Logarithms
The logarithm of a number with an exponent is the product of the exponent and the logarithm of that number.

19. Examples log4(64x5) log464 + log4x5 log464 + 5log4x 3 + 5log4x Expand

20. Examples 6logx + 2logy ln7 – 3lnx logx6 + logy2 ln7 – lnx3 Condense

21. Change-of-Base Property
The logarithm of M (any positive number) with base b is equal to the logarithm of M with any new base divided by the logarithm of b with that new base.

22. Example Use a calculator to evaluate log1571 to four decimal places.
Problem
Solution
Use a calculator to evaluate log1571 to four decimal places.
Apply change of base property.
Let a = 10, b = 15, M = 71
Example

23. Exponential Equations
An exponential equation is an equation containing a variable in an exponent.
Some exponential equations can be solved by expressing each side of the equation as a power of the same base.

24. Example Solve 3x-2 = 9x+4 3x-2 = 32(x+4) = 32x+8
Problem
Solution
Solve 3x-2 = 9x+4
We can rewrite the equation in the form bM = bN and set M = N.
3x-2 = 32(x+4) = 32x+8
x – 2 = 2x + 8
-10 = x or x = -10
Example

25. Solving Exponential Equations
Most exponential equations cannot be rewritten so that each side has the same base.

26. Solving Exponential Equations
Isolate the exponential expression.
Take the natural log on both sides of the equation for bases of other than 10. Use the common log for base 10.
Simplify using one of these properties.
ln bx = x ln b
ln ex = x
log 10x = x
Solve for the variable.

27. Example Solve 5x = 1.4 ln5x = ln1.4 xln5 = ln1.4 x = (ln1.4)/(ln5)
Problem
Solution
Solve 5x = 1.4
Isolate the exponential expression.
Take the natural log on both sides.
Simplify using
ln bx=x ln b or
ln ex = x
ln5x = ln1.4
xln5 = ln1.4
x = (ln1.4)/(ln5)
Example

28. Example Solve 400e0.005x = 1600 400e0.005x = 1600 e0.005x = 4
Problem
Solution
Solve 400e0.005x = 1600
Isolate the exponential expression.
Take the natural log on both sides of the equation.
400e0.005x = 1600
e0.005x = 4
ln e0.005x = ln 4
0.005x = ln 4
x = ln 4/0.005
Example

29. Example Solve e2x – 6ex + 5 = 0. e2x – 6ex + 5 = 0
Problem
Solution
Solve e2x – 6ex + 5 = 0.
This equation is quadratic in form.
It can be factored.
Set each factor equal to zero and solve.
e2x – 6ex + 5 = 0
(ex - 5)(ex – 1) = 0
ex – 5 = 0
ex = 5
ln ex = ln 5
x = ln 5
ex – 1 = 0
ex = 1
x = 0
{0, ln 5}
Example

30. Logarithmic Equations
A logarithmic equation is an equation containing a variable in a logarithmic expression.
Some logarithmic equations can be expressed in the form logbM = c. These can be solved by rewriting in exponential form.

31. Logarithmic Equations
Remember— logarithmic expressions are defined only for logarithms of positive real numbers.
Always check proposed solutions of a logarithmic equation in the original equation.
Exclude any proposed solutions that produce the log of a negative number or the log of zero.

32. Example Solve log6(4x – 1) = 3 Rewrite in exponential form:
Problem
Solution
Solve log6(4x – 1) = 3
Rewrite in exponential form:
63 = 4x – 1
216 = 4x – 1
217 = 4x
54.25 = x
Example

33. Solving Logarithmic Equations
Express the equation in the form logb M = logb N (single logarithm with coefficient of 1.)
Use the one-to-one property to rewrite the equation without logarithms.
If logb M = logb N, then M = N.
Solve for the variable.
Check the proposed solutions in the original equation.

34. Example Solve 2ln(3x) = 8 Remember 2ln (3x) = 8 ln (3x) = 4 e4 = 3x
Problem
Solution
Solve 2ln(3x) = 8
Remember
eln x = x
So eln 3x = 3x
2ln (3x) = 8
ln (3x) = 4
e4 = 3x
e4/3 = x
Example

35. Example Solve logx + log(x + 15) = 2 log(x(x + 15)) = 2
Problem
Solution
Solve logx + log(x + 15) = 2
log(x(x + 15)) = 2
log(x2 + 15x) = 2
102 = x2 + 15x
100 = x2 + 15x
x2 + 15x = 0
(x + 20)(x – 5) = 0
x = 5, -20
{5}
-20 results in a negative log
Example

36. Problem
Solution
Solve ln(x – 4) – ln(x + 1) = ln6
Example

37. Exponential Growth and Decay
F(t) = A = A0ekt
k > 0
Graph rises from left to right.
F(t) = A = A0ekt
k < 0
Graph falls from left to right.
Exponential Growth and Decay

38. Problem
Solution
The function A = 82.3e-0.002t models the population of Germany, A, in millions t years after 2003.
What was the population of Germany in 2003?
A = 82.3e-0.002(0)
A = 82.3e0
A = 82.3 million
Application

39. Problem
Solution
The function A = 82.3e-0.002t models the population of Germany, A, in millions t years after 2003.
Is the population of Germany increasing or decreasing?
Decreasing, since the exponent representing the growth rate is negative.
Application

40. Problem
Solution
The function A = 82.3e-0.002t models the population of Germany, A, in millions t years after 2003.
In what year will the population of Germany be million?
81.5=82.3e-0.002t
0.99 = e-0.002t
ln0.99 = lne-0.002t
-0.002t = ln0.99
2008
Application