1. LOGARITHMS AND EXPONENTIAL MODELS
5.2
LOGARITHMS AND EXPONENTIAL MODELS

2. Using Logarithms to Undo Exponents
Example 2 The US population, P, in millions, is currently growing according to the formula P = 299e0.009t, where t is in years since When is the population predicted to reach 350 million? Solution We want to solve the following equation for t: 299e0.009t = 350. The US population is predicted to reach 350 million during the year 2024.

3. Doubling Time
Example 4 (a)
Find the time needed for the turtle population described by the function P = 175(1.145)t to double its initial size.
Solution (a)
The initial size is 175 turtles; doubling this gives 350 turtles. We need to solve the following equation for t:
175(1.145)t = 350
1.145t = 2
log 1.145t = log 2
t · log = log 2
t = log 2/log ≈ years.
Note that (1.145)5.119 ≈ 350

4. Since the growth rate is 0.000121, the growth rate factor is b = 1 – 0.000121
= Thus after t years the amount left will be

6. Half-Life
Example 8
The quantity, Q, of a substance decays according to the formula Q = Q0e−kt, where t is in minutes. The half-life of the substance is 11 minutes. What is the value of k?
Solution
We know that after 11 minutes, Q = ½ Q0. Thus, solving for k, we get Q0e−k·11 = ½ Q0
e−11k = ½
−11k = ln ½
k = ln ½ / (−11) ≈ ,
so k = per minute. This substance decays at the continuous rate of 6.301% per minute.

7. Converting Between Q = a bt and Q = a ekt
Any exponential function can be written in either of the two forms: Q = a bt or Q = a ekt. If b = ek, so k = ln b, the two formulas represent the same function.

8. Converting Between Q = a bt and Q = a ekt
Example 9
Convert the exponential function P = 175(1.145)t to the form P = aekt.
Solution
The parameter a in both functions represents the initial population. For all t, 175(1.145)t = 175(ek)t,
so we must find k such that ek =
Therefore k is the power of e which gives By the definition of ln, we have k = ln ≈
Therefore, P = 175e0.1354t.
Example 10
Convert the formula Q = 7e0.3t to the form Q = a bt.
Solution
Using the properties of exponents, Q = 7e0.3t = 7(e0.3)t.
Using a calculator, we find e0.3 ≈ , so Q = 7(1.3499)t.

9. Exponential Growth Problems That Cannot Be Solved By Logarithms
Example 13 With t in years, the population of a country (in millions) is given by P = 2(1.02)t, while the food supply (in millions of people that can be fed) is given by N = 4+0.5t. Determine the year in which the country first experiences food shortages. Solution Setting P = N, we get 2(1.02)t = 4+0.5t, which, after dividing by 2 and taking the log of both sides can simplify to t log 1.02 = log( t). We cannot solve this equation algebraically, but we can estimate its solution numerically or graphically: t ≈ So it will be nearly 200 years before shortages are experienced
Finding the intersection of linear and exponential graphs
Population
(millions)
Shortages start →
N = 4+0.5t
P = 2(1.02)t
t, years

10. The growth factor b and the doubling time T are
related by:

11. The growth factor k and the doubling time T are
related by: