# 3.2 Notes (Completed) - Fort Thomas Independent Schools

P.o.D.

1.) If the annual rate of inflation average

4% over the next 10 years, the approximate cost of goods will be modeled by 𝐶(𝑡) = 𝑃(1.04) 𝑡

, where t is the time in years and P is the present cost. The price of an oil change is presently \$23.95. Estimate the price 10 years from now.

2.) Let Q represent a mass of carbon whose half-life is 5715 years. The quantity of carbon-14 present after t 𝑡 years is 𝑄 = 10 (

1

2

)

5715

. Determine the amount present 2000 years later.

1.) 𝐶(10) = 23.95(1.04) 10 = \$35.45

2000

2.) 𝑄 = 10 (

1

2

)

5715

= 7.85 𝑔𝑟𝑎𝑚𝑠

3.2 – Logarithmic Functions and Their

Graphs

Learning Target: be able to evaluate logarithmic functions with base a.

Definition of a Logarithmic Function with Base a: 𝑓(𝑥) = 𝑙𝑜𝑔 𝑎 𝑥 is called the logarithmic function with base a.

Converting Logarithms to Exponential

Equations: 𝑦 = 𝑙𝑜𝑔 𝑎 𝑥 →→→ 𝑥 = 𝑎 𝑦 http://www.youtube.com/watch?v=c-FVjcPpz54

EX: Use the definition of a logarithmic function to evaluate each logarithm at the indicated value of x. a.) 𝑦 = 𝑙𝑜𝑔

4 𝑥, 𝑥 = 16 b.) 𝑦 = 𝑙𝑜𝑔

2 𝑥, 𝑥 = 64 c.) 𝑓(𝑥) = 𝑙𝑜𝑔

5 𝑥, 𝑥 = 1 d.) 𝑓(𝑥) = 𝑙𝑜𝑔

3 𝑥, 𝑥 =

1

81 a.) 𝑦 = 𝑙𝑜𝑔

4

16 → 16 = 4 𝑦

4 2 = 4 𝑦

→ 2 = 𝑦 b.) 𝑦 = 𝑙𝑜𝑔

2

64 → 64 = 2 𝑦

2 6 = 2 𝑦

→ 6 = 𝑦 c.) 𝑦 = 𝑙𝑜𝑔

5

1 → 1 = 5 𝑦

→ 1 = 5 0 → 𝑦 = 0 . Remember, anything to the zero power is 1. d.) 𝑦 = 𝑙𝑜𝑔

3

1

81

1

81

= 3 𝑦 →

3

1

4

= 3 𝑦 → 3 −4 = 3 𝑦 → −4 = 𝑦

Common Logarithms:

-

These are known as “Base 10” logs

Will be written as log(x) or 𝑙𝑜𝑔

10

(𝑥) .

Can be done on the calculator

EX: Use a calculator to evaluate the function given by f(x)=log(x) at each value of x to four decimal places. a.) X=100 b.) X=0.2 c.) X=3.25 d.) X= -4

a.) 2 b.) -0.6990 c.) 0.5119 d.) ERROR – we cannot take the common log of a negative number

*Study the “Properties of Logarithms” on

Pg. 230.

EX: Use the “Properties of Logarithms” to simplify each of the following: a.) 𝑙𝑜𝑔

5

1 b.) 𝑙𝑜𝑔

√11

√11 c.) 8 𝑙𝑜𝑔

8

30 a.) 0 b.) 1 c.) 30

EX: Solve.

a.) 𝑙𝑜𝑔

5 𝑦 = 𝑙𝑜𝑔

5

16 b.) log(4 – 3x) = log (x+2) c.) 𝑙𝑜𝑔

3

(𝑥 2 + 4) = 𝑙𝑜𝑔

3

29 a.) 𝑦 = 16 b.) 4 − 3𝑥 = 𝑥 + 2 →

4 − 2 = 𝑥 + 3𝑥 → c.) 𝑥 2 + 4 = 29

2 = 4𝑥 →

1

= 𝑥

2

→ 𝑥 2 = 25 → 𝑥 = ±5

EX: In the same coordinate plane, sketch the graph of each function. a.) 𝑓(𝑥) = 4 𝑥 b.) 𝑔(𝑥) = 𝑙𝑜𝑔

4 𝑥

a.-b.)

EX: Sketch the graph of 𝑓(𝑥) = log ( 𝑥

4

) .

Identify the vertical asymptote.

Vertical asymptote: x=0.

EX: Describe the graph as a transformation of the graph of 𝑓(𝑥) = 𝑙𝑜𝑔

3 𝑥 . a.) 𝑔(𝑥) = 𝑙𝑜𝑔

3

(𝑥) − 1

b.) ℎ(𝑥) = 𝑙𝑜𝑔

3

(𝑥 + 2) a.) The graph has been shifted 1 unit downward. b.) The graph has been shifted 2 units to the left.

The Natural Logarithmic Function: f(x) = ln (x)

This is the opposite of the Euler function, e .

EX: Use a calculator to evaluate the function given by f(x)=ln(x)+1 for each value of x to four decimal places.

a.

X=73.25 b.

X=0.4 c.

X= -2 d.

𝑥 = 2 + √3 a.

5.2939 b.

0.0837 c.

ERROR – We cannot take the natural logarithm of a negative number. d.

2.3170

*Study and memorize the “Properties of

Natural Logarithms” on the top of page

234.

EX: Use the Properties of Natural

Logarithms to simplify each expression. a.

ln 𝑒

1

3

b.

𝑒 ln 8 c.

15 ln 1 d.

ln 𝑒

6 a.

1/3 b.

8 c.

0 d.

1/6

*Note: we can only take the logarithm

(natural, common, or base) of a positive number.

EX: Find the domain of each function. a.

f(x) = ln(x+3) b.

f(x) = ln(3-x) c.

𝑓(𝑥) = ln 𝑥 3 a.

𝑥 + 3 > 0 → 𝑥 > −3 → (−3, ∞) b.

3 − 𝑥 > 0 → 3 > 𝑥 → (−∞, 3)

c.

𝑥 3 > 0 → 𝑥 > 0 → (0, ∞)

EX: Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given by the human memory model f(t)=78-17log(t+1), 0<t<12, where t is time in months. a.

What was the average score on the original exam (t=0)? b.

After 3 months? c.

After 11 months? a.

78-17log(0+1)=78-17(0)=78-0=78 b.

78-17log(3+1)=78-17log(4)=67.76 c.

78-17log(11+1)=78-17log(12)=

59.65

Upon completion of this lesson, you should be able to:

1.

Convert logarithms to exponential functions and vice versa.

2.

Evaluate common logarithms.

3.

Evaluate natural logarithms.

4.

Explain the difference between common logs and natural logs.