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 written as 10 raised to some power, x. We write this relationship in equation form, like this:
y = 10 x.
For example it is quite obvious that 1000 can be written as 10 3, because the exponent 3 means multiply 10 by itself 3 times and 10·10·10=1000. It may not be quite so obvious that 16 can be written as 10 1.2. What does it mean to multiply something by itself 1.2 times? And how can we calculate that this is the correct power of 10? The answer is in the graph shown below.

This is a graph of the equation  y = 10 x  that was mentioned above. To make this graph we made a table of a few obvious values of  y = 10 x  as shown below, left. Then we plotted the values in the graph (they are the red dots) and drew a smooth curve through them. Then we observed that the curve went through y = 16 and x = 1.2 (the black dot). We take this to mean that 16 = 10 1.2.


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We next define a function called the logarithm that takes a number like 16 as input, calculates that it can be written 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.


Definition: log(x) is defined as 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.




Example 1:   Evaluate log ( 10 5.7 ). In this example the argument of the log function (ie. 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 still has the name “log” but we now include the base, written as a subscript, as part of its name. Here are some examples:

16 = 2 4 log 2 (16) = 4
16 = 4 2 log 4 (16) = 2
16 = 16 1 log 16 (16) = 1
Guided by these examples, we now give the following, more general definition of a logarithm in any base:


Definition: log b(x) is defined as 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.



Notes:


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 (ie. 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
Division property
Exponentiation property

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:



Examples: For each of the following expressions, use the properties of logarithms (or exponents) to combine the logarithms into a single logarithm: Solutions:

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. And 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:
Here is a comparison table for common logarithms and natural logarithms:

Common logarithms Natural logarithms
the base 10 e ≈ 2.718
an example in exponential form y = 10 3 = 1000 y = e 3 ≈ 20.08
the same example in logarithmic form log(y) = 3 ln(y) = 3



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.



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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