This dapper dude is John Napier the "inventor" of logarithms.  If you're curious you can learn more about him by clicking on his picture.  He was a Scot like my distant ancestors (my nostrils flare when I hear bagpipe music!).  The following section explains the concept of expressing numbers in scientific notation.  If you understand scientific notation then it's not too big a stretch to comprehend how logarithms work.  There are a number of self-test problems on this page.  Be sure to attempt these problems before looking at the answers!  The answers are listed at the bottom of the page. If these problems give you no trouble then the quiz should be straightforward.  On the other hand if you can't figure out how to do the problems it's time to find help.

Scientific Notation

To more compactly express and manipulate very large or very small numbers it is convenient to use the concept of scientific notation.  We did some of this in the first classes of the semester. Using this notation a number such as 95,000,000 can be expressed as 9.5x107 or the number 0.00006 can be written 6x10-5 .  Writing numbers in this fashion is based on the use of powers of ten to replace a large number of zeros either before or after the decimal point.  The powers of ten just tells you how many times the number was multiplied or divided by 10.  In the first example above 95 million is written as 9.5 multiplied by 10 seven successive times.  Each multiplication by 10 moves the decimal point along one place so it should be easy to convince yourself that the two representations are equivalent.  The case of 6x10-5  implies that the number 6 is divided by 10 five successive times.  The number that 10 is raised to is called the exponent.  Thus, the exponent in the first example is 7 and in the second example -5.  Remember a negative exponent means divide by 10.  This concept is pretty straightforward; however, in practice folks tend to get confused over the number of zeros.  If you keep in mind the concept of the exponent as the number of times you multiply or divide the original number you should not go wrong.  Try these self test examples.

Self Test 1

Convert the following numbers into scientific notation:

1. 5670000
2. 0.0000546
3. 35

Self Test 2

Convert the following numbers from scientific to regular notation:

1. 3.14 x 107
2. 9.765x10-3
3. 6x1024 (This example makes you appreciate the usefulness of scientific notation!)

One last item to note is that while 101 means multiply by 10, and 10-1 means divide by 10, 100 means multiply by 1!  Actually any number raised to the power 0 equals 1.

Multiplying Numbers in Scientific Notation

How is scientific notation used in operations such as division and multiplication?  Let's examine this simple example: 100 x 6000.  The answer by straightforward multiplication is 600,000.  Now consider the same problem stated in scientific notation: 1 x 102 x 6 x 103  = 6 x 105.  Note that although we multiply the numbers in the normal way (1x6=6), in order to get the appropriate power of 10 we add the exponents (2+3=5).  For the process of division we perform in a similar manner only using subtraction instead of addition.  Again a simple example: 4 x 104 / 2 x 102 = 2 x 102  . Again we divide the regular numbers 4/2 = 2.  For the exponents we subtract the denominator's exponent from that of the numerator 4-2=2.  Of course for the division case you must get the order correct.

Again try these simple examples to make sure that you understand and believe what I've just described.

Self Test 3

1. 103 x 107
2. 2 x 102 x 5 x 10-2
3. 106 / 103
4. 102 x 102 / 105

Exponents and Logarithms

We have seen that when two numbers are multiplied we just add the corresponding exponents.  You should also be getting a vague sense of  how this relation is connected to the logarithmic nature of hearing described in the introductory module in that a multiplicative relation becomes an additive one.  Now we want to introduce the logarithm. The concept of a logarithm is to merely replace a number by the exponent to which 10 would have to be raised to get that number.  For example, consider the number 100.  To what power would 10 have to be raised to get 100?  I hope you cried out 2, since 102 = 100. Thus the logarithm of 100 is 2.  In mathematical terms we would write this as log(100) = 2.  Quick now! What is the logarithm of 1000? 10? 0.1?  If you answered 3, 1, -1 award yourself a major prize.

Now we come to the big leap in understanding.  What is the logarithm of a number like 57?  This number cannot be expressed in a nice easy format of 10 to some integer exponent.  Nevertheless there is still some number that when used as the exponent will give 57.  In the old days to find this number required the use of tables of logarithms.  Now days we use our trusty calculators.  Get your calculator and try these numbers to see for yourself. Not only will this reinforce your understanding--it will also make sure you know where all the appropriate buttons are on your calculator!   Type in 57 and then find the "log" button on your calculator and press it. You should be returned the value 1.75587... This number is the logarithm of 57.  If you think about it this value makes sense because we know that 57 lies between 10 and 100 and the logs of these two numbers are 1 and 2 respectively.

Finally, if we have the log of a number how do we recover the number itself?  Lets continue with the example of 57.  We now know that 1.75587 is the log of 57.  To get from the log to the original number we must use the log value as the exponent of 10.   Some calculators have a 10x button.  In this case enter 1.75587 and hit the 10x button. Voila! You should get 57 (or something pretty close depending on how many significant digits you entered).  Unfortunately not all calculators have the 10x button.  Some require that you enter 1.75587 and then hit INV and then the LOG buttons.  Make sure you figure out how to use your calculator to take the log of a number and to get from the log value back to the number.  You can ask me I've dealt with every manner of calculator over the semesters and I love a challenge.  Try the following self test to verify your calculator savvy.

Self Test 4

1. Find the logs of 50, 500, 5000, 50000.  Do you see a pattern?
2. Find the number whose log is  3.30059548389 (i.e. find the antilog of this value).
3. Find the log of 81.  Divide this log value by 2 and then find the antilog of this value.  How are the initial and final numbers related?
4. Try finding the logs of 1, 0, and -1.

The Rules of Logarithms

Now that you understand what a logarithm means I want to show you a few simple mathematical manipulations that can be done using logarithms.  These are often called the Rules of Logarithms, however they should not be mysterious to you given what we have covered in the previous sections. The first rule is typically expressed as

log(XY) = log X + log Y

In words this rule states that if I take two numbers X and Y and multiply them together and then take the logarithm I obtain the same result as if I had added the logarithms of the two numbers separately.  Consider the simple example of 102 times 103.  Multiplied together these come to 105 and the log of 105 is 5 (that is what the left hand side of the equation tells us to do).  Now the log of 102 is 2 and the right hand side of the equation instructs us to add this to the log of 103 which is 3 to get 5.  The rule works!  Of course this relation is nothing more than putting together the two ideas (1) that the log of a number is the exponent of 10 required to equal that number and (2) when we multiply powers of 10 we add the exponents.

The second rule of logarithms is a straightforward extension of the first--it states that

log (Xn) = n log X

that is, if the number X is raised to the power n the result is the same as n times log of X.  To understand this rule imagine that n=2.  In that case

log (X2) = log (XX) = log (X) + log (X) = 2 log X

where I merely wrote out X2 as XX in the first step and then used the first rule above.  The extension to higher n should be obvious--try writing it out for n=3.

Finally there is a rule for division as well as for multiplication.  This rule states that

log(X/Y) = log X - log Y.

Using these rules one can do some nifty manipulation to reduce the need for calculation.  I know that in these days of calculators this may not seem a big deal but it's still important in understanding the manipulation of logarithms.  For example, in the previous sections self test I asked you to find the logs of a series starting 50, 500 etc.  You should have seen the pattern that resulted and now using Rule 1 you can understand why this pattern came about.  The log of 50 is 1.69897.  If I want the log of 500 and I don't have access to a calculator I can use rule 1 by recognizing that 500 = 50 x 10.  Thus,

log(500) = log(50x10) = log(50) + log(10) = log(50) + 1 = 2.69897

making use of the fact that we know that the log of 10 is 1.  The following self test--the last in this module--will allow you to practice the use of some of the rules of logarithms.  There are hints located a few steps below the questions--just the other side of the talking Punchy.  No calculators for this section!!!

Self Test 5

1. Given that log(2)=0.301 and log(4)=0.602 find log(8).
2. Using the information in the previous question find log(16).
3. Find the log of the square root of 2.
4. Find the log of 400.

Hints to the last self test

1. Use Rule 1 with X=2 and Y=4.
2. Remember 42 =16.
3. Remember square root of 2 squared is equal to 2.  Look to Rule 2!
4. Look at the example at the end of the section.

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