Logarithms

Algebraic Rules and Graphing

Contents

  • Rules for Working with Logarithms

  • Some Aspects of Logarithms

  • Graph of the Logarithm Function: Y = Log (X)


     

    Rules for Working with Logarithms

    The rules for working with logarithms are relatively straight-forward and follow from the definition of the logarithmic function.

    The rules for logarithmic operations that you need to be familiar with are as follows:

    Log(a × b) = Log(a) + Log(b)
    Log(a ÷ b ) = Log(a) - Log(b)
    Log(a) + Log(b) = Log(a) + Log(b) ; not much you can do with this!
    Log(a) - Log(b) = Log(a) - Log(b) ; you can't do much here either!
    Log(ab) = b × Log(a)

    Though these rules are not proven here they follow rather simply from the definition of logarithms. To save space we won't bother with the proofs. For one thing you can test out these rules on your calculator to check and convince yourself. For instance, try the following:

    Log(2 × 3) = Log(2) + Log(3) ?? Is this true?

    Well, Log(2 × 3) = Log(6) = 0.77815, so this is the numerical value for the left hand side. Now, Log(2) = 0.30103 and Log(3) = 0.47712, for the right hand side. Adding the values for the right hand side together we get: Log(2) + Log(3) = 0.77815, which is the same as Log(6). Obviously, this does not prove the relation, but perhaps it helps to convince you that the rules are not random and are based on a definition for the logarithm that is consistent with the rules of algebraic powers.

     

    Some Aspects of Logarithms

    The definition of the logarithmic function, as you saw in section 8.1, leads to some interesting results that might seem unexpected at first glance.

    Log(1) = 0
    For one thing, if you take the logarithm of 1 on your calculator (try this!) you will get a value of zero!! Whay is this so? Again, just like the rules for working with logarithms, this has to do with the way that the logarithm is defined. Recall that when you take the Log base 10 of a number, such as 1, you are asking the following question: what number do I have to raise 10 to the power of in order to get the value of 1? Well, by defninition 10 to the power of 0 is equal to 1, so that is why the Log(1) =0! In fact, since any number - except 0 - raised to the power of 0 gives 1, then the logarithm of the value of 1 will always be zero no matter what base you are working in. Thus, the natural logarithm of 1 is also equal to zero: Ln(1) = 0. Try this out on your calculator too!

    Negative Answers When Taking Log

    Now try taking the logarithm of a number that is less than one but greater than zero; for example, try taking Log(0.5). You will notice that you end up with a negative answer, -0.301029995. So, why is the logarithm of 0.5 a negative number? Again, if one considers the definition of logarithms this makes sense. You are asking what number do I have to raise 10 to the power of in order to get 0.5? If you take 100 you will get 1 and if you take 101 you will get 10. Thus, in order to get a value less than one you'll have to take 10 to the power of some number that is less than zero, thus 10 to the power of a negative number! So, don't be surprised when you take the Log of a number and you end up with a negative answer. If the number you are taking the logarithm of is less than one you will get a negative number out.

    Log(0) = undefined!!!!
    Finally, try taking the Log(0) on your calculator. This is perhaps going to annoy you and you might even start thinking that your calculator is broken! The Log(0) will result in the calculator giving you an error message, meaning that what you are trying to do is not a good operation. To understand this result, imagine you start taking the logarithm of numbers less than one, like you did above for Log(0.5), but keep taking the Log of smaller and smaller numbers: try Log(0.1), Log(0.01), Log(0.001), which should give you -1, -2, -3 respectively. Everytime you will get a smaller negative number. Well, if you were to iterate this process until you got very close to zero you would notice that you are getting very large negative number. You'll probably run out of patience before you get anywhere, but as you approach taking the Log of very very small numbers you will end up with extremely large negative numbers, until the function becomes undefined because you have gone too far close to zero! So, the Log(0) is undefined and besides this you don't need to know more about it at this level. In fact, you should only take the Log of numbers that are greater than zero. The logarithmic function that we have introduced here is undefined for zero or for numbers less than zero (i.e., Log(-1) also gives you an error).

    Graph of the Logarithm Function

    To get a larger picture of what the logarithmic function does we can take the function y = Log(x) and plot it on an x-y graph. If you don't feel comfortable graphing functions you should probably review the section on graphing functions. Basically, to plot y = Log(x), one can make a table of x values and take the logarithm of each of these values and create a table of y values. This is shown below:

    table of x values: (0.1, 0.5, 1, 2, 5, 10, 20)
    table of y values: (-1, -0.303, 0, 0.303, 0.699, 1, 1.3)

    (Note: each value on the y-table is equal to the Log of the corresponding value in the x-table ... thus -1=Log(0.1), -0.303=Log(0.5), etc ...)

    We can then plot an x-y graph, by taking pairs of values from each table and plotting each one as a point on an x-y grid. This yields the following graph: