The logarithm as an inverse function
In this section we concentrate on understanding the logarithm function. If
the logarithm is understood as the inverse of the exponential function,
then the properties of logarithms will naturally follow from our
understanding of exponents.
Elementary Functions
Part 3, Exponential Functions & Logarithms
Lecture 3.3a, Logarithms: Basic Properties
The meaning of the logarithm.
The logarithmic function g(x) = logb (x) is the inverse of the exponential
function f (x) = bx .
Dr. Ken W. Smith
The meaning of y = logb (x) is by = x.
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The expression
by = x
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is the “exponential form” for the logarithm y = logb (x).
The positive constant b is called the base (of the logarithm.)
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Logarithms
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Logarithms
Some worked exercises.
Write each of the following logarithms in exponential form and then use
that exponential form to solve for x.
4
1
log2 (8) = x
Solution. The exponential form is 2x = 8. Since 23 = 8 the answer is
x=3.
2
log2 (247 ) = x
Solution. The exponential form is 2x = 247 . So x = 47 .
3
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log2 ( 21 ) = x
Solution. The exponential form is 2x = 21 . Since 2−1 =
1
the answer
2
5
log2 ( 18 ) = x
Solution. The exponential form is 2x = 18 . Since 2−3 = 81 the answer
is x = −3 .
√
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!
is x = −1 .
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The graph of a logarithm function
The graph of a logarithm function
The graph of y = 2x was drawn in an earlier lecture (see below.)
The graph of y = log2 x :
The graph of the inverse function y = log2 x is obtained by reflecting the
graph of y = 2x across the line y = x.
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The graph of a logarithm function
The graph of a logarithm function
If we draw them together, we have the picture below.
The graph of the exponential function y = 2x :
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The graph
of the logarithmic function
y = log2 x:
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Logarithms
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 log
function will be the range of the exponential function, and vice versa.
In summary, here are our abbreviations:
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) by
ln(x) and speak of the “natural logarithm”.
1
ln x means the logarithm base e,
2
log x means the logarithm base 10 and
3
lb x means the logarithm base 2.
Sometimes, for historical reasons, we may use base 10. It is customary to
speak then of the “common logarithm” and abbreviate log10 (x) by log(x),
dropping the subscript. However (warning!), in higher mathematics and
engineering applications, log(x) usually means base e and is equivalent to
ln(x).
In these notes we will use log(x) to mean log10 (x).
One more abbreviation – in computer science, because computers store
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data in binary (in bits of zeroes and ones), one uses base 2. Some
abbreviate log2 (x) as lb(x) and speak of the “binary” logarithm.
Logarithms
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2
3
log(1000) = x
Solution. The exponential form is 10x = 1000. Since 103 = 1000 the
answer is x = 3 .
1
ln( 3 ) = x
e
Solution. The exponential form is ex = e−3 so the answer is −3 .
1
lb( √ ) = x
2
√
1
Solution. The exponential form is 2x = √ . Since 21/2 = 2 then
2
1
−1/2
2
= √ and so the answer is x = −1/2 .
2
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Properties of exponential functions in terms of logarithms
A few more worked exercises.
Write each of the following logarithms in exponential form and then use
that exponential form to solve for x.
1
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The logarithm function plucks the exponent from an expression. For this
reason, the properties of exponents translate into properties of logarithms.
For example, we know that when we multiply two terms with a common
base, 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 the
product M N ?
The answer is “We add the exponents appearing in M and N .” In other
words (if we learn to translate “logb ” as “the exponent on b that...”), we
can restate this exponent property as “when we multiply numbers we add
their exponents”. This is the product property for logarithms:
logb (M N ) = logb M + logb N
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Logarithms
Logarithms
What happens when we divide two terms with a common base?
A third important property of exponents: when we raise a term like bx to a
power, we multiply exponents.
bx
= bx−y
by
(3)
When we do division, we subtract exponents. So, in the language of
logarithms, we have the quotient property, “the exponent in a quotient is
the difference of the two exponents”:
Elementary Functions
(5)
In our “logarithm language” (thinking of M as bx ) we have the exponent
property
logb (M c ) = c logb M
M
logb ( ) = logb M − logb N
N
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(bx )c = bxc
(6)
(4)
Each of these three properties is merely a restatement, in the language of
logarithms, of a property of exponents.
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Logarithms
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Exponential Functions
We review the three basic logarithm rules we have developed so far.
Product Property of Logarithms:
logb (M N ) = logb M + logb N
In the next presentation, we develop several more properties of logarithms.
Quotient Property of Logarithms:
logb (
(END)
M
) = logb M − logb N
N
Exponent Property of Logarithms:
logb (M c ) = c logb M
Each of these properties is a restatement, in the language of logarithms, of
a property of exponents.
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Logarithms
We review the three basic logarithm rules we have developed so far.
Product Property of Logarithms:
logb (M N ) = logb M + logb N
Elementary Functions
Part 3, Exponential Functions & Logarithms
Lecture 3.3b, Logarithms: Basic Properties, Continued
Quotient Property of Logarithms:
logb (
Dr. Ken W. Smith
M
) = logb M − logb N
N
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Exponent Property of Logarithms:
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logb (M c ) = c logb M
Each of these three properties is merely a restatement of a property of
exponents.
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Changing the base
Changing the base
Suppose we want to change the base of our logarithm. This often occurs
when we want to use a “good” base like e on a problem which began with
a different base.
Suppose we want to work with base c but our problem began with base b:
We began with
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y = logb x.
We rewrote this as
y=
y = logb x.
logc x
.
logc b
So y, which was originally equal to logb x is now
Rewrite this in exponential form:
by = x.
Now take the log of both sides of the equation. If we want to work in base
c then let us apply logc () to both sides of our equation.
y
logc (b ) = logc (x).
Now we use the exponent property pulling the exponent y outside the
logarithm:
y logc (b) = logc (x).
log x
c
Elementary Functions
y=
.
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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.
x
Then on the right side of the equal sign ( log
log b ), b is still lower than x!
Solve for y:
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Logarithms
More on the logarithm as an inverse function
We began this lecture by defining g(x) = logb (x) as the inverse function of
f (x) = bx . Since these functions are inverses, we know then that
(f ◦ g)(x) = (g ◦ f )(x) = x.
Example. Suppose we want to compute log2 (17) but our calculator only
allows us to use the natural logarithm ln. Then, by the change of base
equation we can write
log2 (17) =
ln 17
≈ 4.087463.
ln 2
(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 to
just 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 (bx ) as
“the exponent on b which gives bx .”
Obviously this should be x since x is the exponent one places on b to get
bx . (If that doesn’t make sense, read through it one more time slowly....)
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More on the logarithm as an inverse function
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Six properties of logarithms
In summary, we have developed the following six properties of logarithms.
1
The Product Property of Logarithms:
Since (f ◦ g)(x) = x we also have
x = (f ◦ g)(x) = f (g(x)) = f (logb x) = blogb x . So
logb (M N ) = logb M + logb N
blogb x = x.
2
The Quotient Property of Logarithms:
This is almost as easy to understand as the previous equation.
logb (
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!
3
M
) = logb M − logb N
N
The Exponent Property of Logarithms:
logb (M c ) = c logb M
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Six properties of logarithms
4
More on the logarithm as an inverse function
The Change of Base Property
logb x =
5
If we understand the logarithm as the inverse of the exponential function
then we are prepared to find the inverse of a variety of functions. Here are
some examples.
Find the inverse function of:
2
1 f (x) = ex .
2
2 f (x) = ex −5 .
3 f (x) = 5 + ex .
4 f (x) = log2 (x + 2) + 2.
logc x
logc b
Inverse Property #1
logb bx = x
6
Solutions
2
2
1 To find the inverse of f (x) = ex set y = ex and swap inputs and
outputs
2
x = ey .
Inverse Property #2
blogb x = x
Take the natural logarithm of both sides
ln x = y 2
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More on the logarithm as an inverse function
and solve for y by taking square roots of both sides
√
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Elementary
lnFunctions
x = y.
√
−1 (x) =
onethe
inverse
function isasfan
ln xfunction
.
MoreSoon
logarithm
inverse
4
2
2
ex −5
To find the inverse of y =
take natural logs of both sides
we swap letters so that x =
2
ey −5 ,
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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 and
outputs)
x = log2 (y + 2) + 2,
2
ln x = y − 5,
add 5 and take square roots so that
and subtract 2 from both sides
√
f −1 (x) = ln x + 5 .
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x − 2 = log2 (y + 2).
To find the inverse of y = 5 + ex we swap variables, subtract 5 from
both sides and then take the natural log to get ln(x − 5) = y. So
At this point it is best to write this logarithmic equation in
exponential form.
2x−2 = y + 2.
Subtract 2 from both sides
2x−2 − 2 = y
f −1 (x) = ln(x − 5) .
and then write out our answer using inverse function notation.
f −1 (x) = 2x−2 − 2
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Exponential Functions
In the next series of lectures, we apply properties of logarithms.
(END)
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