12.2 - Logarithms
Before reading this section you may want to review the section on
exponents, since logarithms are based on them.
Introduction: It is a fact that every positive number, y,
can be expressed as 10 raised to some
power, x. We write this relationship in equation form, like this:
y = 10 x
For example it is obvious that the number 1000 can be expressed as 10 3,
because the exponent 3 means to multiply 10 by itself 3 times and
10·10·10=1000.
It is not so obvious that the number 16 can be expressed as 10 1.2.
Clearly this cannot mean to multiply 10 by itself 1.2 times! What does it mean and how can we calculate
that this is the correct power of 10? We start by giving it a name: logarithm. We say that 1.2 is the logarithm of 16. It is the subject of this chapter.
Let’s start by making a graph of the equation y = 10 x.
To make this graph we make a table of a few obvious values of
y = 10 x as shown below, left.
Then we plot the values in the graph (they are the red dots) and draw a smooth
curve through them. Then we observe that the curve goes through y = 16
and x = 1.2 (the black dot). This is what we mean when we say that 16 = 10 1.2.
We next define a function called the logarithm function that takes a number like 16
as input, calculates that it can be expressed as 10 1.2
(click here to see exactly how this is done by your calculator),
and returns the exponent 1.2 as its output value:
Here is the formal definition of the logarithm function.
Definition: The logarithm is the function that takes
any positive number x as input and returns the
exponent to which the base 10 must be raised to obtain x.
It is denoted log(x).
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Example 1: Evaluate log ( 10 5.7 ).
In this example the argument of the
log function (i.e. the quantity in brackets) is already expressed
as 10 raised to an exponent, so the log function simply returns the exponent.
log ( 10 5.7 ) = 5.7
Example 2: Evaluate log ( 1000 ). The argument is a number that is easily expressed as 10
raised to an exponent. We do this and the log function then returns the exponent.
log ( 1000 ) = log ( 10 3 ) = 3
Example 3: Evaluate log ( 16 ). The argument is a number
which we don't know how to express as 10 raised to an exponent
(unless we remember the above discussion which said that 16 = 10 1.2 ).
Therefore we use a calculator or the Algebra Coach to evaluate it.
log ( 16 ) = 1.2
Example 4: log ( x + 4 ). The argument is an expression.
Until we can evaluate that expression we have no choice but to leave this logarithm as is.
Note that the number 16 can be expressed in exponential form using various bases,
so various types of logarithms can be defined. Each type of logarithm is still denoted ‘log’ but we now include the base, written as a subscript, as part of the notation.
Here are some examples:
| 16 = 2 4
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→
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log 2 (16) = 4
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| 16 = 4 2
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→
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log 4 (16) = 2
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| 16 = 16 1
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→
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log 16 (16) = 1
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Guided by these examples, we now give the following, more general definition of a logarithm in any base:
Definition: The logarithm to base b is the function
that takes any positive number x as input and returns the
exponent to which the base b must be raised to obtain x.
We denote it as logb(x).
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Notes:
- If the subscript after the word ‘log’ is omitted then the base of the logarithm is understood to be 10.
Thus log10(x) and log(x) are the same thing.
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Suppose that x and y are related by the equation
x = b y.
This says that ‘x is b raised to the exponent y’.
This is called the exponential form of this relationship.
But this same relationship can also be written as
log b(x) = y,
This says that ‘the logarithm of x to base b is y’.
This is called the logarithmic form of this relationship.
These two equations are equivalent.
They are like saying ‘Mary is John’s mother’ and
‘John is Mary’s son’.
We say that we are ‘taking logs’ when converting from exponential
to logarithmic form, and ‘taking antilogs’ when converting from
logarithmic to exponential form.
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If we substitute the exponential form x = b y into its own
logarithmic form log b(x) = y we get the identity
log b( b y ) = y,
and if we substitute the logarithmic form log b(x) = y into
its own exponential form b y = x we get the identity
These identities show how the inverse operations of logging and antilogging undo each other.
Example: For our previous example 16 = 10 1.2 the above two
identities read
log10(10 1.2 ) = 1.2,
and
Example: Evaluate each of the following logarithms without using a calculator:
Solution: The key is to express the argument of the log
function (i.e. the quantity in brackets) in exponential form with the base chosen to match the base of the log.
Then we use the fact that taking logs and exponentiation are inverse operations:
In the chapter on exponents we stated these three
properties of exponents:
| Multiplication property
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| Division property
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| Exponentiation property
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If we rewrite them in logarithmic form then they become the properties of logarithms.
To do this, make these substitutions on the left side of each of the three properties:
b m = x and b n = y
Note for later reference that
these substitutions expressed in logarithmic form are:
m = log b(x) and n = log b(y).
With the substitutions the three properties of exponents read:
Now take logs of these three equations
(i.e. write them in logarithmic form):
Now substitute log b(x) for m and
log b(y) for n
on the right side of each property (but only for m in the third one).
The result is three properties of logarithms.
The properties of logarithms are:
- Property 1: the logarithm of a product:
- Property 2: the logarithm of a quotient:
- Property 3: the logarithm of an exponential:
These properties are very useful for simplifying a logarithm or for
combining several logarithms into one logarithm.
Another useful property can be gotten by letting x = 1 in property 2 or by
letting m = −1 in property 3:
- Property 4: the logarithm of a reciprocal:
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Examples: For each of the following expressions, use the properties of logarithms (or
exponents) to combine the logarithms into a single logarithm:
Solutions:
Step 1: get rid of the coefficient 3 by using property 3 to make it
an exponent of 3;
Step 2: combine the sum of logs using property 1:
Step 1: get rid of the coefficients 2 and 5 by using property 3;
Step 2: combine the difference of logs using property 2:
Step 1: combine the sum of the first two logs using property 1;
Step 2: combine the difference of the remaining logs using property 2;
This could be handled exactly the same way as the previous problem but a
shortcut is to notice that all positive logs go into the numerator and all negative logs
go into the denominator.
Step 1: get rid of the coefficients 3 and 6 by using property 3;
Step 2: combine the sum of logs using property 1;
Step 3: write a 3 b 6 as
(a b 2 ) 3.
Click here to see why you can;
Step 4; use property 3 of logarithms again, but this time in reverse:
A shortcut is to notice that a common factor of 3 can be factored out of the two
terms at the very beginning:
Common logarithms and natural logarithms
Suppose that we wish to express an arbitrary positive number y in exponential
form y = b x. The base b that we use could be any positive
number whatsoever except 0 or 1. This is because 0x can only equal 0 and
1x can only equal 1 for any value of x. Also, if we try a negative b
then we run into trouble with b1/2 since this is the square root of
a negative number. Of all the remaining possibilities for the base b there are two
special values:
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base b = 10 This would seem to be the obvious choice since our
number system is based on 10 (probably due to the fact that humans
have 10 fingers with which they first learned to count!)
Logarithms to base 10 are called common logarithms.
For convenience we omit the subscript 10 when using common logs.
Thus log(x) is understood to mean log10(x).
All scientific calculators are programmed to compute logarithms to base 10
(on most calculators you use the log button) and antilogs to base 10
(use the 10 x button.)
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base b = e ≈ 2.71828…
This may not seem like a natural choice for the base but nevertheless logarithms to base
e are called natural logarithms. The importance of base e
(which is the symbol for an irrational number whose value is approximately equal
to 2.718) results from the fact that the exponential growth function
y = e x, with that particular base,
is the only function whose slope equals its own height everywhere.
This makes it important in the study of any quantity whose rate of growth is
proportional to its present value. (An example is the balance in an interest
bearing bank account.)
Click here for more information on the function
y = e x.
For convenience we will use the abbreviation ln(x) instead of the longer
form log e(x) to represent the natural logarithm
function. (LN stands for Log Natural.)
All scientific calculators are programmed to compute logarithms to base e
(on most calculators use the ln button) and antilogs to base e (use the
e x button.)
Here is a comparison table for common logarithms and natural logarithms:
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Common logarithms
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Natural logarithms
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| the base
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10
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e ≈ 2.718
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| an example in exponential form
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y = 10 3 = 1000
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y = e 3 ≈ 20.08
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| the same example in logarithmic form
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log(y) = 3
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ln(y) = 3
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The Change of Base Formulas
Logarithms to base 10 and logarithms to base e are proportional to each other
(just like miles and kilometers). Logarithms in one base can be changed to logarithms in any other
base using a ‘change of base formula’ which basically just uses a proportionality constant.
There is also a change of base formula that can be applied to a relationship
expressed in exponential form.
Logarithmic form of the Change of Base Formula:
Exponential form of the Change of Base Formula:
These formulae are used to convert from an inconvenient base c to a convenient
base b (usually 10 or e). The left side is usually converted to the
right side. The quantity
log b(c) is the
conversion factor.
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Proof: To prove the second formula notice that
it is just the identity discussed previously, namely
, with x replaced
by c. To prove the first formula, start with the second formula in the form:
and follow these steps:
Example: Compute log 3 (8).
Solution: Use the logarithmic form of the change of base formula:
Example: Express the function y = 7 x using the
base e instead of the base 7.
Solution: Use the exponential form of the change of base formula on the number 7.
Remember that ln(7) means the same thing as log e(7).
7 = e ln (7) = e 1.946
Substitute this expression in for 7 in y = 7 x.
y = (7) x =
(e 1.946 ) x =
e 1.946 x
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