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Transcript of Chapter 3.1-3.3
Exponential and Logarithimic Functions
World Life Uses
Logarithms are used in both nuclear and internal medicine. For example, they are used for investigating pH concentrations, determining amounts of radioactive decay, as well as amounts of bacterial growth. Logarithms also are used in obstetrics. When a woman becomes pregnant, she produces a hormone known as human chorionic gonadotropin. Since the levels of this hormone increase exponentially, and at different rates with each woman, logarithms can be used to determine when pregnancy occurred and to predict fetus growth.
3.1 Exponential Functions and Graphs Vocabulary
Natural base e
The irrational number e ≈ 2.718281828 . . .
Increasing the number of compounding in the compound interest formula without bound leads to continuous compounding, which is given by the formula A = Pert.
The exponential function
f with base a is denoted by f(x) = a^x , where a > 0, a ≠1, and x is any real number
natural exponential function
is given by the function f(x) = e^x . In this function, e is the constant and x is the variable.
A one-to-one function
the property that if a and b are in the domain of f then f(a)=f(b) if and only if a=b
Archaeologists use logarithms to determine the age of artifacts, such as bones and other fibers, up to 50,000 years old. When a plant or animal dies, the isotope of carbon Carbon-14, it decays into the atmosphere. Using logs, archaeologists can compare the decaying Carbon-14 to the Carbon-12, which remains constant in an organism even after death, to determine the age of the artifact. For example, this type of carbon dating was used to determine the age of the Dead Sea Scrolls.
When a population has a constant growth rate, its size can be calculated using a natural exponential function. The population P after t units of time P(t) = P(0)e kt , where k is the constant relative growth rate, and P(0) is the initial population, measure at time zero. The units of time used in problems like these usually are proportional to the life span of the organisms of the population. For populations of bacteria, hours or days are common, and for people, years are common. Populations can also be shrinking. In shrinking populations the k would be negative but everything else would remain the same
Exponential functions can be used to model the concentration of a drug in a patient's body. The concentration of Drug X in a patient's bloodstream is modeled by,
C (t) = C0 e - rt,
where C (t) represents the concentration at time t (in hours), C0 is the concentration of the drug in the blood immediately after injection, and r > 0 is a constant indicating the removal of the drug by the body through metabolism and/or excretion. The rate constant r has units of 1/time (1/hr).
logarithms that use 10 as the base
change of base formula :
for all positive numbers a, b, and n, where a does not equal 1 and b does not equal 1, log b n= log b n/log b a