TeacherExcellenceWorkshop

July 20, 2008July 20, 2008

Logarithmicand Exponential

Functions

D. P. Dwiggins, PhDDepartment of

Mathematical Sciences

TeacherExcellenceWorkshop

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

Rules for Exponentsx y x ya a a

0 1a 1xx

aa

x

x yy

aa

a

2 2*xxa a

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.

Rules for Exponentsx y x ye e e

0 1e 1xx

ee

x

x yy

ee

e

22*x xe e

Exponential functions turn sums intoproducts, so that additive properties become multiplicative properties.

The additive identity (zero) becomesthe multiplicative identity (one).

The additive inverse (negative) becomesthe multiplicative inverse (reciprocal).

Inverse addition (subtraction) becomesinverse multiplication (division).

Multiplication becomes exponentiation.

Turn Sums Into Products

( ) ( ) ( )E x y E x E y Let E(x) denote any function which turns sums into products:

Exponential Functions

Setting y = 0 into the above equation gives

( 0) ( ) (0)E x E x E which means

( ) (0) ( )E x E E x must be true for every x.

But the only way for this to be true is if

(0) 1E Therefore exponential functions turn the additive identity

into the multiplicative identity.

Turn Sums Into Products

( ) ( ) ( )E x y E x E y Let E(x) denote any function which turns sums into products:

Exponential Functions

Setting y = –x into the above equation gives

( ( )) ( ) ( )E x x E x E x which means

( ) ( ) (0)E x E x E But E(0) = 1, and so E(x) and E(–x) must be

multiplicative reciprocals. That is,

1( ) ( )E x E x Exponential functions turn additive inverses into multiplicative inverses.

Turn Sums Into Products

( ) ( ) ( )E x y E x E y Let E(x) denote any function which turns sums into products:

Exponential Functions

Setting y = x into the above equation gives

( ) ( ) ( )E x x E x E x which means

2(2 ) ( )E x E x

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.

of Exponential Functions

( )If ( ) ( ) then 1

( )

( )* ( ) ( ) 1

0

E xE x E y

E y

E x E y E x y

x y y x

Every exponential function is one-to-one:

Logarithms Are Inverses

Thus, every exponential function E(x) has an inverse,and the inverse of an exponential function defines

a logarithmic function L(x),

where ( ) means ( ) .y L x E y x In terms of a specific base (a) for the exponential

function, this statement becomeslog ( ) means .y

ay x a x

Turn Products Into Sums

( ) ( ) ( )L x y L x L y Let L(x) denote any function which turns products into sums:

Logarithmic Functions

Setting y = 1 into the above equation gives

( 1) ( ) (1)L x L x L which means

( ) (1) ( )L x L L x must be true for every x.

But the only way for this to be true is if

(1) 0L Therefore logarithmic functions turn the multiplicative

identity into the additive identity.

Turn Products Into Sums

( ) ( ) ( )L x y L x L y Let L(x) denote any function which turns products into sums:

Logarithmic Functions

Setting y = 0 into the above equation gives

( 0) ( ) (0)L x L x L which means

( ) (0) (0)L x L L must be true for every x.

(0) is undefinedL

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 thelogarithmic function nontrivial, we have to assume

Turn Products Into Sums

( ) ( ) ( )L x y L x L y Let L(x) denote any function which turns products into sums:

Logarithmic Functions

Setting y = 1/x into the above equation gives

( (1/ )) ( ) (1/ )L x x L x L x which means

( ) (1/ ) (1)L x L x L But L(1) = 0, and so L(x) and L(1/x) must be

negatives of each other. That is,

(1/ ) ( )L x L xExponential functions turn multiplicative inverses into additive inverses.

Turn Products Into Sums

( ) ( ) ( )L x y L x L y Let L(x) denote any function which turns products into sums:

Logarithmic Functions

Setting y = x into the above equation gives

( ) ( ) ( )L x x L x L x which means

2 2* ( )L x L xTaking 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.

Exponential Function

0 0

( ) ( ) ( ) ( ) ( ) 1

( ) (0)( ) , and taking 0 gives

the derivative ( ), where (0).

E E x x E x E x E x E x

x x xE x E

E x xx

dEm E x m E

dx

Let E(x) be any exponential function. To calculate itsderivative, we begin by calculating the difference quotient:

The Derivative of an

In other words, the derivative of any exponential functionis equal to itself, multiplied by a constant scale factor,

and this constant is the slope of the tangent line at (0,1).

Exponential FunctionThe exponential function which has the property that the

slope of the tangent line at (0,1) has the value m0 = 1is called the natural exponential function, written asexp(x) or more frequently as ex, where e is the base ofthe exponential function which has unit slope at (0,1).

The Derivative of an

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.

x xd

dxe e

ex has the property of being equal to its own derivative:

Logarithmic FunctionLet y = L(x), corresponding to x = E(y).

Since the derivative of E is equal to itself times m0,

The Derivative of a

In particular, when m0 = 1, so that L(x) becomes thenatural logarithmic function ln(x), we have

0 0

0 0

we have ( )

1or

dx m E y dy m xdy

dx dydy

m x dx m x

as the derivative of the logarithmic function L(x).

1ln

dx

dx x

Since L(x) is defined for x > 0, the above shows L always hasa positive derivative, and so logarithms are strictly increasing.

Logarithmic FunctionThe Derivative of a

we can use logarithms to calculate definite integrals wherethe integrand is a reciprocal function. For example,

1Because ln ,

dx

dx x

Textbooks then start with this and derive all the properties oflogarithmic functions, which is completely backwards, sincewe have just shown that this “definition” follows from the

algebraic properties of exponential and logarithmic functions.

11

1ln ln ln1 ln for any 0.

t tdx x t t tx

Many textbooks turn this equation around, switching x and t,

and write a “definition” of the natural logarithm as

1

1ln , which is valid for all 0.

xx dt x

t

Population GrowthApplications:

, where is called the growth constant.dP

k P kdt

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 changesis proportional to the size of P. In terms of differentials,

Rearranging this equation gives , which thendP

k dtP

gives the integral equation .dP

kdtP

This last integral is just kt,

while the first integral is logarithmic. Including a constant ofintegration gives lnP + C = kt, where the constant C is evaluatedusing the initial condition: P = P0 at t = 0 C = – lnP0. Thus,

0 00 0ln ln , or ln , which gives , or .kt ktP P

P PP P kt kt e P P e

Radioactive DecayApplications:

0ktdQ

k Q Q Q edt

The population growth model P = P0ekt represents a quantitywhich 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 thedecay constant, and this model represents radioactive decay.

Usually in this type of application the decay constant k isnot given; instead, what is given is the time that it takes forhalf of the initial quantity to decay. ( is called the half-life.)

01 1

2 2ln 2

at 2 .k kQ Q t e e k

0

1Using this, the decay model can be rewritten as .

2tQ Q

Newton’s Law of Cooling

Applications:

Another application of the decay model is to consider thequantity Q = – R, where is the temperature of a hotobject which is cooling down to room temperature, R.

Problems utilizing all three types of these models are

given on the next slide.

R 0 RThen becomes ( ) ,ktdQ

k Q edt

where 0 is the object’s original temperature at time t = 0.

ProblemsApplications:

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?

# 1.

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?

# 2.

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.)

# 3.

Regression Analysis

Given a collection of data,

0102030405060708090100

0 1 2 3 4 5 6 7 8 9 10 11

Regression Analysis

Impose a linear relation on the data points:

0102030405060708090100

0 1 2 3 4 5 6 7 8 9 10 11

Regression AnalysisFind an equation for the line by “regressing” the data points back to the imposed linear relation:

0102030405060708090100

0 1 2 3 4 5 6 7 8 9 10 11

Least Squares Method( )i i iAX B Y Let

between the predicted value for Yi and theactual value (i is also called the residual).

denote the error

Let S denote a measure of the total error betweenthe data points and the regression line.

1

N

ii

S

will not work, as the positive errors will cancel the negative errors and this sum will always equal zero.

1

N

ii

S

is a possibility, but awkward to use because the absolute value function is non-differentiable at its local minimum.

Least Squares MethodLeast Squares Regression finds the equationfor the regression line by minimizing thesum of the squares of the errors:

2 2

1 1

( )N N

i i ii i

S AX B Y

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.