Elementary Functions The logarithm as an inverse...

Elementary FunctionsPart 3, Exponential Functions & LogarithmsLecture 3.3a, Logarithms: Basic Properties

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 29

The logarithm as an inverse function

In this section we concentrate on understanding the logarithm function. Ifthe logarithm is understood as the inverse of the exponential function,then the properties of logarithms will naturally follow from ourunderstanding of exponents.

The meaning of the logarithm.The logarithmic function g(x) = logb(x) is the inverse of the exponentialfunction f(x) = bx.

The meaning of y = logb(x) is by = x.

The expressionby = x

is the “exponential form” for the logarithm y = logb(x).

The positive constant b is called the base (of the logarithm.)


Logarithms

Some worked exercises.Write each of the following logarithms in exponential form and then usethat exponential form to solve for x.

1 log2(8) = xSolution. The exponential form is 2x = 8. Since 23 = 8 the answer isx = 3 .

2 log2(247) = x

Solution. The exponential form is 2x = 247. So x = 47 .

3 log2(12) = x

Solution. The exponential form is 2x = 12 . Since 2−1 =

1

2the answer

is x = −1 .


Logarithms

4 log2(18) = x

Solution. The exponential form is 2x = 18 . Since 2−3 = 1

8 the answer

is x = −3 .

5 log2(3√2) = x

Solution. The exponential form is 2x = 3√2 = 21/3. So x = 1/3 .

Notice how we use the exponential form in each problem!


The graph of a logarithm function

The graph of y = 2x was drawn in an earlier lecture (see below.)

The graph of the inverse function y = log2 x is obtained by reflecting thegraph of y = 2x across the line y = x.



The graph of y = log2 x :



If we draw them together, we have the picture below.



The graph of the exponential function y = 2x:

The graph of the logarithmic function y = log2 x:Smith (SHSU) Elementary Functions 2013 8 / 29

Logarithms

We agreed earlier that the exponential function f(x) = bx has domain(−∞,∞) and range (0,∞).

Since g(x) = logb x is the inverse function of f(x) the domain of the logfunction will be the range of the exponential function, and vice versa.

So the domain of logb x is (0,∞) and the range is (−∞,∞).

The most useful base for logarithms is e. We will abbreviate loge(x) byln(x) and speak of the “natural logarithm”.

Sometimes, for historical reasons, we may use base 10. It is customary tospeak then of the “common logarithm” and abbreviate log10(x) by log(x),dropping the subscript. However (warning!), in higher mathematics andengineering applications, log(x) usually means base e and is equivalent toln(x).

In these notes we will use log(x) to mean log10(x).

One more abbreviation – in computer science, because computers storedata in binary (in bits of zeroes and ones), one uses base 2. Someabbreviate log2(x) as lb(x) and speak of the “binary” logarithm.


Logarithms

In summary, here are our abbreviations:

1 lnx means the logarithm base e,

2 log x means the logarithm base 10 and

3 lb x means the logarithm base 2.


Logarithms

A few more worked exercises.Write each of the following logarithms in exponential form and then usethat exponential form to solve for x.

1 log(1000) = xSolution. The exponential form is 10x = 1000. Since 103 = 1000 theanswer is x = 3 .

2 ln(1

e3) = x

Solution. The exponential form is ex = e−3 so the answer is −3 .

3 lb(1√2) = x

Solution. The exponential form is 2x =1√2. Since 21/2 =

√2 then

2−1/2 =1√2

and so the answer is x = −1/2 .


Properties of exponential functions in terms of logarithms

The logarithm function plucks the exponent from an expression. For thisreason, the properties of exponents translate into properties of logarithms.For example, we know that when we multiply two terms with a commonbase, we add the exponents:

(bx)(by) = bx+y (1)

Suppose we call the first term M := bx and the second term N := by.Then one may ask the question, “What is the exponent on b in theproduct MN?The answer is “We add the exponents appearing in M and N .” In otherwords (if we learn to translate “logb” as “the exponent on b that...”), wecan restate this exponent property as “when we multiply numbers we addtheir exponents”. This is the product property for logarithms:

logb(MN) = logbM + logbN (2)


http://en.wikipedia.org/wiki/Binary_logarithm

Logarithms

What happens when we divide two terms with a common base?

bx

by= bx−y (3)

When we do division, we subtract exponents. So, in the language oflogarithms, we have the quotient property, “the exponent in a quotient isthe difference of the two exponents”:

logb(M

N) = logbM − logbN (4)


Logarithms

A third important property of exponents: when we raise a term like bx to apower, we multiply exponents.

(bx)c = bxc (5)

In our “logarithm language” (thinking of M as bx) we have the exponentproperty

logb(Mc) = c logbM (6)

Each of these three properties is merely a restatement, in the language oflogarithms, of a property of exponents.


Logarithms

We review the three basic logarithm rules we have developed so far.

Product Property of Logarithms:

logb(MN) = logbM + logbN

Quotient Property of Logarithms:

logb(M

N) = logbM − logbN

Exponent Property of Logarithms:

logb(Mc) = c logbM

Each of these properties is a restatement, in the language of logarithms, ofa property of exponents.


Exponential Functions

In the next presentation, we develop several more properties of logarithms.

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Elementary FunctionsPart 3, Exponential Functions & Logarithms

Lecture 3.3b, Logarithms: Basic Properties, Continued

Dr. Ken W. Smith

Sam Houston State University

2013


Logarithms

We review the three basic logarithm rules we have developed so far.

Product Property of Logarithms:


Quotient Property of Logarithms:

logb(M


Exponent Property of Logarithms:

logb(Mc) = c logbM

Each of these three properties is merely a restatement of a property ofexponents.


Changing the base

Suppose we want to change the base of our logarithm. This often occurswhen we want to use a “good” base like e on a problem which began witha different base.Suppose we want to work with base c but our problem began with base b:

y = logb x.

Rewrite this in exponential form:

by = x.

Now take the log of both sides of the equation. If we want to work in basec then let us apply logc() to both sides of our equation.

logc(by) = logc(x).

Now we use the exponent property pulling the exponent y outside thelogarithm:

y logc(b) = logc(x).

Solve for y:

y =logc x

logc b.Smith (SHSU) Elementary Functions 2013 19 / 29

Changing the base

We began withy = logb x.

We rewrote this as

y =logc x

logc b.

So y, which was originally equal to logb x is now

logb x =logc x

logc b

Let’s call this the “change of base” equation or “change of base”property.

One way to remember this is to note that on the left side of the equal sign(logb x), b is lower than x.Then on the right side of the equal sign ( log xlog b ), b is still lower than x!


Logarithms

Example. Suppose we want to compute log2(17) but our calculator onlyallows us to use the natural logarithm ln. Then, by the change of baseequation we can write

log2(17) =ln 17

ln 2≈ 4.087463.


More on the logarithm as an inverse function

We began this lecture by defining g(x) = logb(x) as the inverse function off(x) = bx. Since these functions are inverses, we know then that

(f ◦ g)(x) = (g ◦ f)(x) = x. (7)

Let us examine this in more detail.

Note that (g ◦ f)(x) = g(f(x)) = g(bx) = logb(bx). Since the log function

and the exponential function are inverse functions, this must be equal tojust x and we have

logb(bx) = x.

This equation is really fairly easy to understand. If we translate “logb(x)”as “the exponent on b that give x” then we should translate logb(b

x) as“the exponent on b which gives bx.”Obviously this should be x since x is the exponent one places on b to getbx. (If that doesn’t make sense, read through it one more time slowly....)



Since (f ◦ g)(x) = x we also havex = (f ◦ g)(x) = f(g(x)) = f(logb x) = blogb x. So

blogb x = x.

This is almost as easy to understand as the previous equation.

It says that if we place on b “the exponent you put on b to get x” (logb x)then we should just get x!


Six properties of logarithms

In summary, we have developed the following six properties of logarithms.

1 The Product Property of Logarithms:


2 The Quotient Property of Logarithms:

logb(M


3 The Exponent Property of Logarithms:

logb(Mc) = c logbM


Six properties of logarithms

4 The Change of Base Property

logb x =logc x

logc b

5 Inverse Property #1logb b

x = x

6 Inverse Property #2

blogb x = x



If we understand the logarithm as the inverse of the exponential functionthen we are prepared to find the inverse of a variety of functions. Here aresome examples.Find the inverse function of:

1 f(x) = ex2.

2 f(x) = ex2−5.

3 f(x) = 5 + ex.4 f(x) = log2(x+ 2) + 2.

Solutions

1 To find the inverse of f(x) = ex2

set y = ex2

and swap inputs andoutputs

x = ey2.

Take the natural logarithm of both sides

lnx = y2

and solve for y by taking square roots of both sides√lnx = y.

So one inverse function is f−1(x) =√lnx .



2 To find the inverse of y = ex2−5 we swap letters so that x = ey

2−5,take natural logs of both sides

lnx = y2 − 5,

add 5 and take square roots so that

f−1(x) =√lnx+ 5 .

3 To find the inverse of y = 5 + ex we swap variables, subtract 5 fromboth sides and then take the natural log to get ln(x− 5) = y. So

f−1(x) = ln(x− 5) .



4 To find the inverse of f(x) = log2(x+ 2) + 2 we write

y = log2(x+ 2) + 2,

change variables (to indicate that we are swapping inputs andoutputs)

x = log2(y + 2) + 2,

and subtract 2 from both sides

x− 2 = log2(y + 2).

At this point it is best to write this logarithmic equation inexponential form.

2x−2 = y + 2.

Subtract 2 from both sides

2x−2 − 2 = y

and then write out our answer using inverse function notation.

f−1(x) = 2x−2 − 2Smith (SHSU) Elementary Functions 2013 28 / 29

Exponential Functions

In the next series of lectures, we apply properties of logarithms.

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