# 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

2.) 𝑄 = 10 ( 1 2 ) 2000 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. 1 d.) 𝑦 = 𝑙𝑜𝑔 3 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. For more information, visit http://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.a

spx HW Pg.236 6-84 6ths, 88, 107, 108