Section 6.3 – Exponential FunctionsLaws of Exponents

If s, t, a, and b are real numbers where a > 0 and b > 0, then:

𝑎𝑠 ∙𝑎𝑡=𝑎𝑠+𝑡 (𝑎𝑠)𝑡=𝑎𝑠𝑡 (𝑎𝑏)𝑡=𝑎𝑡𝑏𝑡

1𝑡=1 𝑎0=1 𝑎−𝑡= 1𝑎𝑡=( 1

𝑎 )𝑡

Definition:

An Exponential Function is in the form,

“a” is a positive real number and does not equal 1

“C” is a real number and does not equal 0

The domain of f(x) is the set of all real numbers

“a” is the base and is the Growth Factor

“C” is the Initial Value because

𝑓 (𝑥+1)𝑓 (𝑥 )

=𝑎→𝐶𝑎𝑥+1

𝐶𝑎𝑥 =𝐶𝑎𝑥𝑎1

𝐶𝑎𝑥 =𝑎

Section 6.3 – Exponential FunctionsExamples

𝑓 (0 )=1 , h𝑡 𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝐶=1

𝑥 𝑓 (𝑥) 𝑓 (𝑥+1)𝑓 (𝑥 )

=𝑎

−1

0

1

2

3

23

132

94

278

12/3

=32

3/21

=32

9/ 43 /2

=32

27/ 89/4

=32

𝑎=32

𝑓 (𝑥 )=𝐶𝑎𝑥

𝑓 (𝑥 )=1( 32 )

𝑥

¿ ( 32 )

𝑥

Section 6.3 – Exponential FunctionsExamples

𝑓 (0 )=1/ 4 , h𝑡 𝑒𝑟𝑒𝑓𝑜𝑟𝑒𝐶=1/4

𝑥 𝑓 (𝑥) 𝑓 (𝑥+1)𝑓 (𝑥 )

=𝑎

−1

0

1

2

3

12

14

18

116

132

1/41/2

=12

1/81/4

=12

1/161/8

=12

1/321/16

=12

𝑎=12

𝑓 (𝑥 )=𝐶𝑎𝑥

𝑓 (𝑥 )=14 ( 3

2 )𝑥

Section 6.3 – Exponential FunctionsProperties of the Exponential Function,

The domain is the set of all real numbers.

The range is the set of all positive real numbers.

The y-intercept is 1; x-intercepts do not exist.

The x-axis (y = 0) is a horizontal asymptote, as x.If a > 1, the f(x) is increasing function.

The graph contains the points (0, 1), (1, a), and (-1, 1/a).

The graph is smooth and continuous.

Section 6.3 – Exponential Functions

𝑓 (𝑥 )=𝑎𝑥 ,𝑎>1The graph of the exponential function is shown below.

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a

Section 6.3 – Exponential FunctionsProperties of the Exponential Function,

The domain is the set of all real numbers.

The range is the set of all positive real numbers.

The y-intercept is 1; x-intercepts do not exist.

The x-axis (y = 0) is a horizontal asymptote, as x.If 0 < a < 1, then f(x) is a decreasing function.

The graph contains the points (0, 1), (1, a), and (-1, 1/a).

The graph is smooth and continuous.

Section 6.3 – Exponential Functions

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1The graph of the exponential function is shown below.

𝑓 (𝑥 )=𝑎𝑥 ,0<𝑎<1

Section 6.3 – Exponential FunctionsEuler’s Constant – e

The value of the following expression approaches e,

(1+ 1𝑛 )

𝑛

as n approaches .

Using calculus notation,

Growth and decay

Compound interest

Differential and Integral calculus with exponential functions

𝑒= lim𝑛→∞ (1+ 1

𝑛 )𝑛

Applications of e

Infinite series

Section 6.3 – Exponential Functions

Solving Exponential Equations

1)

2)

2 𝑥=5

𝑥=52

3)

2 𝑥+4=3

𝑥=−12

Theorem

If , then .

Section 6.3 – Exponential FunctionsSolving Exponential Equations

4)

5)

−4 𝑥=3

𝑥=−34

34 ∙3−4 𝑥− 4=33

Section 6.4 – Logarithmic FunctionsThe exponential and logarithmic functions are inverses of each other.

The logarithmic function is defined by

𝑦=𝑙𝑜𝑔𝑎𝑥𝑖𝑓 𝑎𝑛𝑑𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥=𝑎𝑦

The domain is the set of all positive real numbers .

The range is the set of all real numbers .

The x-intercept is 1 and the y-intercept does not exist.

The y-axis (x = 0) is a vertical asymptote.

If 0 < a < 1, then the logarithmic function is a decreasing function.

The graph contains the points (1, 0), (a, 1), and (1/a, –1).

The graph is smooth and continuous.

If a > 1, then the logarithmic function is an increasing function.

Section 6.4 – Logarithmic Functions

(𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

The graph of the logarithmic function is shown below.

The natural logarithmic function

𝑦=𝑙𝑜𝑔𝑒𝑥=ln 𝑥𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥=𝑒𝑦

The common logarithmic function

𝑦=𝑙𝑜𝑔10 𝑥=𝑙𝑜𝑔𝑥 𝑖𝑓 𝑎𝑛𝑑𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥=10𝑦

Section 6.4 – Logarithmic Functions

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a

(𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

Graphs of

Section 6.4 – Logarithmic Functions

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a (𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

Graphs of

Inverse Functions:

Section 6.4 – Logarithmic FunctionsGraphs of

Inverse Functions:

Section 6.4 – Logarithmic Functions

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a (𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

Graph

Section 6.4 – Logarithmic Functions

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a (𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

Graph

Section 6.4 – Logarithmic Functions

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a (𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

Graph

Section 6.4 – Logarithmic Functions

𝑦= 𝑓 (𝑥 )=𝑎𝑥

a (𝑎 ,1 )

𝑎

𝑦=𝑙𝑜𝑔𝑎𝑥

Graph

Section 6.4 – Logarithmic FunctionsChange the exponential statements to logarithmic statements

𝑎5=6.75=𝑙𝑜𝑔𝑎6.7

8𝑥=9.2𝑥=𝑙𝑜𝑔8 6.7

𝑒3=𝑏3=ln𝑏

𝑒𝑥=4𝑥=ln 4

Change the logarithmic statements to exponential statements

5=3𝑥

𝑙𝑜𝑔3 5=𝑥7=𝑥4

𝑙𝑜𝑔𝑥7=4

𝑎=𝑒6

ln 𝑎=6

Solve the following equations𝑙𝑜𝑔3 (2𝑥 )=1

2 𝑥=31

𝑥=32

𝑙𝑜𝑔2(5 𝑥+1)=4

5 𝑥+1=24

5 𝑥+1=165 𝑥=15𝑥=3

𝑙𝑜𝑔2 (32 )=−3 𝑥+9

32=2−3 𝑥+9

5=−3 𝑥+9−4=−3𝑥43=𝑥

25=2−3 𝑥+9

Section 6.4 – Logarithmic FunctionsSolve the following equations

𝑒𝑥=10𝑥=ln 10

𝑒7𝑥=157 𝑥=ln15

𝑥=ln 15

7

8+2𝑒𝑥=12

𝑥=ln 2𝑒𝑥=2

2𝑒𝑥=4