Chapter 6 - Logarithms - Applications

Question 1

What do you do when given A (t) = A₀eᵏᵗ, k < 0? (5)

Answer 1

• Law of uninhibited decay for half life
• A (t) = Amount remaining at (t)
• A₀ = Initial Amount (when t = 0)
• k = Decay Constant (negative)
• Initially solve for k first, then plug back into equation to solve for other variable

Question 2

How do you initially solve for when given A (t) = A₀eᵏᵗ, k < 0? (4)

Answer 2

a. Replace A (t) w/ 1/2 A₀
b. Replace t w/ half-life amount
c. Eliminate A₀ on both sides
d. Switch to logarithmic form to solve for k

Question 3

In half-life applications, what equation do you use if not already given one?

Answer 3

If outside 1 year range: A (t) = A₀ (1/2) ᵗ/ʰ
If many times per year: A (t) = A₀ (1/2) ᵗʰ

Question 4

What is the logistic growth model? (2)
Ex: P (t) = 0.90/1 + 3.5e⁻⁰.³³⁹ᵗ

Answer 4

• Numerator refers to maximum amount possible
• Whenever they give you a number, just plug into P (t) as it is

Question 5

What equation do you use when they ask if something doubles, triples, etc. (3)

Answer 5

1. A (t) = A₀ (2) ᵗ/ʰ , A (t) = A₀ (3) ᵗ/ʰ , etc.
2. Time will not always be years, can infer it could be in other units of time
3. If they just mention that it doubles every 7 hours, t is in hours)

Question 6

What do you do if you are given an unfamiliar equation in exponential/logarithmic applications?

Answer 6

• They will give you enough info, like what each variable represents
• If there is a “k” variable, solve for k FIRST

Question 7

How do you solve Newton’s Law of cooling? (2)
Given this info:
u (t) = T + (u₀ - T)eᵏᵗ, k < 0

u (t) is temperature at a given time (t)
T is the constant temperature
u₀ is the initial temperature

Answer 7

1. T is room/air temperature
2. Solve for k FIRST (plug in everything else & switch to log form)

Question 8

How do you find exponential regression?

Answer 8

1. Must beginning year as “0”, then add on with same intervals
2. USE CALCULATOR: Stat, Edit, Plug in L1 & L2, Stat, Calc, 0
3. Plug a and b into y = a (b)ˣ
4. y = population/cost/etc. –> x = time (years) –> a = initial population (when x = 0)
   –>b = growth factor

Question 9

Growth Factor vs Growth Rate

Answer 9

Growth Factor: (1 + r)
Growth Rate: r