1. Logarithmic and Exponential Functions
20 July 08
Logarithmic
and Exponential
Functions
Teacher Excellence Workshop
D. P. Dwiggins, PhD
Department of
Mathematical Sciences
Teacher Excellence Workshop
July 20, 2008
July 20, 2008

2. Goals and Activities Properties of Exponential Functions
Properties of Logarithmic Functions
As Inverses of Exponential Functions
Logarithmic and Exponential Graphs
Logarithm as Area
Slopes of Tangent Lines for Logarithmic and Exponential Curves
Definition of the Natural Exponential Base
Applications
Population Growth Models
Radioactive Decay
Newton’s Law of Cooling

3. Rules for Exponents
In this list of rules, a denotes a positive real number, called the base of the exponential function.
We could use a particular base (such as 2 or 10), and we can choose any letter to stand for the base.
In the next slide, the base will be denoted by the letter e.

4. Rules for Exponents
Exponential functions turn sums into products, so that additive properties become multiplicative properties.
The additive identity (zero) becomes the multiplicative identity (one).
The additive inverse (negative) becomes the multiplicative inverse (reciprocal).
Inverse addition (subtraction) becomes inverse multiplication (division).
Multiplication becomes exponentiation.

5. Exponential Functions Turn Sums Into Products
Let E(x) denote any function which turns sums into products:
Setting y = 0 into the above equation gives
which means
must be true for every x. But the only way for this to be true is if
Therefore exponential functions turn the additive identity into the multiplicative identity.

6. Exponential Functions Turn Sums Into Products
Let E(x) denote any function which turns sums into products:
Setting y = –x into the above equation gives
which means
But E(0) = 1, and so E(x) and E(–x) must be multiplicative reciprocals. That is,
Exponential functions turn additive inverses into multiplicative inverses.

7. Exponential Functions Turn Sums Into Products
Let E(x) denote any function which turns sums into products:
Setting y = x into the above equation gives
which means
Exponentiation increases the order of arithmetic, turning simple operations into more complicated ones. Exponential functions turn addition into multiplication, and so they also turn multiplication into exponentiation.

8. Logarithms Are Inverses of Exponential Functions
Every exponential function is one-to-one:
Thus, every exponential function E(x) has an inverse, and the inverse of an exponential function defines a logarithmic function L(x),
In terms of a specific base (a) for the exponential function, this statement becomes

9. Logarithmic Functions Turn Products Into Sums
Let L(x) denote any function which turns products into sums:
Setting y = 1 into the above equation gives
which means
must be true for every x. But the only way for this to be true is if
Therefore logarithmic functions turn the multiplicative identity into the additive identity.

10. Logarithmic Functions Turn Products Into Sums
Let L(x) denote any function which turns products into sums:
Setting y = 0 into the above equation gives
which means
must be true for every x.
But the only way for this to be true (if L(0) is defined) is if L(x) = 0 for every x. Thus, in order to make the logarithmic function nontrivial, we have to assume

11. Logarithmic Functions Turn Products Into Sums
Let L(x) denote any function which turns products into sums:
Setting y = 1/x into the above equation gives
which means
But L(1) = 0, and so L(x) and L(1/x) must be negatives of each other. That is,
Exponential functions turn multiplicative inverses into additive inverses.

12. Logarithmic Functions Turn Products Into Sums
Let L(x) denote any function which turns products into sums:
Setting y = x into the above equation gives
which means
Taking logarithms decreases the order of arithmetic, turning complicated operations into simpler ones. Logarithmic functions turn multiplication into addition, and so they also take away exponents and turn them into products.

13. The Derivative of an Exponential Function
Let E(x) be any exponential function. To calculate its derivative, we begin by calculating the difference quotient:
In other words, the derivative of any exponential function is equal to itself, multiplied by a constant scale factor, and this constant is the slope of the tangent line at (0,1).

14. ex has the property of being equal to its own derivative:
The Derivative of an
Exponential Function
The exponential function which has the property that the slope of the tangent line at (0,1) has the value m0 = 1 is called the natural exponential function, written as exp(x) or more frequently as ex, where e is the base of the exponential function which has unit slope at (0,1).
ex has the property of being equal to its own derivative:
This means y = ex solves the differential equation y = y. In general any exponential function solves a diff. eq. of the form y = ky, where k again is the slope of the tangent line at (0,1). Using the chain rule from calculus, we can show any such function is of the form y = ekx.

15. as the derivative of the logarithmic function L(x).
The Derivative of a
Logarithmic Function
Let y = L(x), corresponding to x = E(y). Since the derivative of E is equal to itself times m0,
as the derivative of the logarithmic function L(x).
In particular, when m0 = 1, so that L(x) becomes the natural logarithmic function ln(x), we have
Since L(x) is defined for x > 0, the above shows L always has a positive derivative, and so logarithms are strictly increasing.

16. The Derivative of a Logarithmic Function
we can use logarithms to calculate definite integrals where the integrand is a reciprocal function. For example,
Many textbooks turn this equation around, switching x and t, and write a “definition” of the natural logarithm as
Textbooks then start with this and derive all the properties of logarithmic functions, which is completely backwards, since we have just shown that this “definition” follows from the algebraic properties of exponential and logarithmic functions.

17. Applications: Population Growth
Suppose P(t) represents the size of a growing population P, with initial size P = P0 at t = 0. The basic assumption of a population growth model is that the rate at which P changes is proportional to the size of P. In terms of differentials,
This last integral is just kt,
while the first integral is logarithmic. Including a constant of integration gives lnP + C = kt, where the constant C is evaluated using the initial condition: P = P0 at t = 0  C = – lnP0. Thus,

18. Applications: Radioactive Decay
The population growth model P = P0ekt represents a quantity which grows exponentially with time for any positive k > 0. If k < 0, this model represents a quantity which gets smaller exponentially in time. In this case, k (or –k) is called the decay constant, and this model represents radioactive decay.
Usually in this type of application the decay constant k is not given; instead, what is given is the time t that it takes for half of the initial quantity to decay. (t is called the half-life.)

19. Newton’s Law of Cooling
Applications:
Newton’s Law of Cooling
Another application of the decay model is to consider the quantity Q = Q – QR, where Q is the temperature of a hot object which is cooling down to room temperature, QR.
where Q0 is the object’s original temperature at time t = 0.
Problems utilizing all three types of these models are given on the next slide.

20. Applications: Problems
# 1.
Suppose an initial population of 5,000 moves into an area, and it has a growth rate which causes it to double every ten years. What will the population be 25 years after it initially moved in? When will the population reach 25,000?
# 2.
The radioactive isotope found in corbomite has a half-life of 5,000 years. An artifact containing corbomite is discovered, and after measurement it is found to contain only 10% of the isotope as would be expected upon comparison with a modern sample of corbomite. How old is the artifact?
# 3.
A freshly made cup of coffee has an initial temperature of 90ºC. After one minute, the temperature has cooled down to 60ºC. If the room temperature is 20ºC, how long will it take for the coffee to cool down to body temperature? (Body temperature = 37ºC.)

23. Regression Analysis
Given a collection of data,

24. Regression Analysis
Impose a linear relation on the data points:

25. Regression Analysis
Find an equation for the line by “regressing” the data points back to the imposed linear relation:

26. Least Squares Method Let between the predicted value for Yi and the
actual value (ei is also called the residual).
denote the error
Let S denote a measure of the total error between
the data points and the regression line.
will not work, as the positive errors will cancel the negative errors and this sum will always equal zero.
is a possibility, but awkward to use because the absolute value function is non-differentiable at its local minimum.

27. Least Squares Method Least Squares Regression finds the equation
for the regression line by minimizing the
sum of the squares of the errors:
S is minimized by taking derivatives of S with respect to A and B, setting these derivatives equal to zero, and solving the resulting equations for A and B.