The Power in Numbers
A Logarithms Refresher

UNH Mathematics Center

Spring 2001

• Logarithms are defined in such a ''backwards'' way that sometimes students see only the rules for using them, and can't get a good picture of what they are.

• Tables of logarithms were originally constructed to make calculations easier. And of course everybody uses calculators for this purpose now! So it often happens that students find themselves in a university calculus course without much prior experience with logarithms.

It's not hard to see why the definition of a logarithm has this backwards-looking flavor: the log functions are, after all, inverses to important exponential functions. But this is the very reason they are still indispensable!

They are the functions we need, (in combination with their companion exponential functions) to describe exponential growth.

Every situation in which the growth-rate of a quantity is proportional to its present level is described by an exponential function.

Example: Although a country's birth rate is affected by other factors as well, it will be proportional to the country's present population. Every year, there are more babies born in New York City than in Durham, New Hampshire. This is true of other kinds of ''populations'' as well: dollars, or fish, or bugs, or radioactivity. The logarithm and exponential functions describe them all.

1  Why use two systems of logarithms?

The really odd thing is, that nature loves logarithms too (and Mother Nature isn't biased in favor of the number 10). When you get a bit further into calculus you will see that the definite integral

ó õ x 1  1 t  dt

(which is a function of its variable upper limit x) has all the properties a logarithm function ought to have. It is the logarithm function that Nature provides. Its corresponding exponential function is exp(x), or e^x; and its base is the ''natural number'' e @ 2.718.

The 10-based ''common'' logarithms and the natural logarithms follow the same rules. Their values are even proportional to each other.

It would actually be possible to construct a system of logarithms based on any positive number. Most people figure, the two we have will do nicely.

2  Defining logarithms

We'll start with the common, 10-based logarithms. The idea is to think of each number as a power of 10:

Of course, we don't have many numbers in our table yet! But you can already see a few things:

• When we multiply numbers, their logs add up. Notice, for example, that the log of 100×10,000 = 1,000,000 and log 100 + log 10000 = log 1000000 (that is, 2 + 4 = 6).
• When we divide numbers, the logs are subtracted. For instance, the log of 100/0.01 = 10,000 is 2 - (-2) = 4.
• If we square a number, its log doubles. For instance, the log of (10³)² is 6.
• We only have logs for positive numbers. Some of the logs are negative: they are the logs of numbers smaller than 1.

The first tables of logarithms were constructed by John Napier, working in his castle at Merchiston in Scotland. It took him a good 20 years, and it's all the more remarkable because he didn't even have exponential notation to work with.

Nevertheless Napier's ''wonderful reckoning numbers'' are exponents. There are other ways of calculating logarithms now, and by the time you study power series you'll see how it can be done. For the time being, try out a few of them with your calculator. You should be aware the logarithms are real numbers, which we can only approximate using decimals or other fractions. So it's only to be expected that the decimal numbers you'll see require some rounding before you recognize them as the numbers they really are!

• The common logarithm of 3 is (to five figures, at least) 0.47712. Try it on your calculator. Find the tens-powers key, and ask the calculator for 10^0.47712.
• Notice that on your calculator, the tens-powers function and its inverse log function share the same key.
• Use the logarithms key on your calculator to get log 8. To five figures, it should be 0.90309. That tells us that 10^0.90309 @ 8. You can use either the tens-powers key or the ordinary powers key (often labeled with an up-arrow) to find the power 10^0.90309. Try it. you should get 8.
• The sum of 0.90309 and 0.47712 is 1.38021. What number has 1.38021 as its logarithm? A number between 10¹ and 10², we think - because its logarithm is between 1 and 2. Ask your calculator for 10^1.38021, and see if you don't get 24 = 8·3. When we add logs of two numbers, we get the log of their product.

• And while we're about it, ask your calculator for the common logarithm of 2.4. You should get (within rounding!) 0.38021, the fractional part of log(24).

• The common logarithms go well with decimal numbers. If we know that log 3.75 = 0.57403 then we also know that
  • log 0.375 = log(3.75×10^-1) = 0.57403-1
  • log 37.5 = log(3.75×10¹) = 1.57403
  • log 37,500 = log(3.75×10⁴) = 4.57403.

  The fractional part of the logarithm, 0.57403, gives us the significant digits 3.75 and its integer part tells us the power of 10 that multiplies the 3.75.

• If we divide 3 by 8, we get 0.375. So the logarithm of 3/8 should be 0.47712-0.90309 = -0.42597. Although a calculator doesn't do it, it's been customary to write this number as 0.57403-1, because 0.57403 is the log of 3.75.
• There's no good way to find log 11 from log 8 and log 3. It's not their sum!
• One-third of 0.90309 (log 8) is 0.30103. Ask your calculator for 10^0.30103. It should return 2, which is the cube root of 8. When we divide a number's logarithm by 3, we have the logarithm of the number's cube root.

2.1  A note about notation

One time you must use parentheses is if the argument to a logarithm function is a sum. If you want the logarithm of the number 2x+5, you must write ''log(2x+5).'' If you omit the parentheses and write ''log 2x+5'' everyone who reads your work will think you meant ''(log 2x) + 5'' or '' 5 + log 2x.'' Unfortunately, this will include the person who reads your quiz papers!

2.2  The calculus-based logarithmic function

Its inverse exponential function is ''exp'' and its value-variable is called either ''exp(x)'' or ''e^x.'' You can pronounce it either as ''exp'' or ''e-to-the-x.'' The first notation, exp(x), reminds us that the number x is an argument to the function: the second notation, e^x, reminds us of the rules of exponents that the function's values follow.

The number e itself, which is exp(1) or e¹, is an irrational number: 2.718 is only an approximation to its value. It is fine to use the name e instead of the approximate decimal value: e is, in fact, the correct name for this particular number.

Leonhard Euler (pronounced ''oiler'') was the key figure in 18th century mathematics, and you will guess correctly that the numerical base of the natural logarithms is still called e in his honor.

All the ''rules'' of logarithms are the same, whether we use 10 as a base (common logarithms) or e as a base (natural logarithms). The systems are even proportional!

3  Here are the Rules.

4  Using the logarithm and exponential functions

The key to using these functions is to remember how good they are at undoing each other:

Example: The equation

y = 6ln x + 1

gives y ''in terms of x'' - meaning that y is written as an expression whose only variable is x.

Suppose we want to revise it, so that x is given in terms of y. (That is, we want to solve the equation for x.) We begin by solving it for ln x:

ln x = y-1 6

Now we apply the natural exponential function (we choose the natural exponential function instead of the tens-power one, because it's the companion function to the natural logarithm):

exp(ln x ) = x = exp( y-1 6 )

Example: Using a log function is the way to get ''at'' a variable that's part of an exponent:

y = 50·102x.

Let's say we need to find the value of x when y = 450.

We would first solve for the exponential expression:

102x = y 50 = 450 50 = 9

Now we would use a logarithm function to undo the effect of the tens-power exponential function. Either the common or the natural log function will work, but here it seems in the spirit of things to use the common logarithm:

log(102x) = 2x = log(9)

We also know something about the logarithm of a square (it's on the third line of the section on The Rules); and because 9 is the square of 3 we know that log 9 = log 3² = 2log 3.

2x = log(9) = 2log(3) = 2(0.47712)

x = 0.47712

Example: Suppose that the quantity q(t) of some substance depends on the time, t, and that the dependence is observed to be

q(t) = 400 e0.15t

with the time measured in hours. This is a very typical ''model'' of exponential growth, so it's good to be familiar with it.

• At ''time zero'' (when the discussion begins) the quantity of the substance was q(0) = 400·e⁰ = 400. We apparently started with 400 units of the substance.
• Because the argument 0.15t to the exponential function is positive (usually in this model we think of time as going forward, so that t ³ 0) the quantity q(t) will increase. The positive exponent-coefficient 0.15 tells us that this an exponential model describes growth.
• An exponential growth model is often described by its time of doubling. How long will it take in this instance? We need to know the time t for which q(t) = 800:
  

  800 = q(t) = 400 e0.15t

  so that
  

  e0.15t = 2.

  To solve e^0.15t = 2, we would apply a log function. The natural log function seems most appropriate:
  

  ln(e0.15t) = 0.15t = ln(2)

  We look up the log's value: ln(2) @ 0.693. Then,
  

  t = ln(2) 0.15 @ 0.693 .15 = 4.62

  The substance will have doubled in 4.62 hours (about 4 hours 37 minutes).

  And remember what we said earlier about approximations: although logarithms are real numbers we are using decimal numbers to approximate them. So we may be a minute or two off here. What's a minute among friends, after all?

  The doubling continues, every 4.62 hours: at the end of 9.24 hours (about 9 hours 22 minutes) we will have four times the quantity we began with.

5  Some Problems

1. Use the (abbreviated) table of common logarithm values

   number	common log
   2	0.30103
   3	0.47712
   5	0.69897
   7	0.84510
   9	0.95424
   11	1.04139

   to evaluate:
   • log 60
   • log 2/55
   • log 3500
   • log 5.5
   • log 77x² (Your answer will have a ''log x'' term in it.)
   • log (Öx)/6.3

2. Use the properties of the logarithm function to break apart the expression ln[(504x²)/((y+1)³)] into its simplest components. (That includes factoring 504! It's divisible by 8.)

3. In our example of exponential growth, someone in some lab somewhere must have observed the substance carefully enough to come up with the model q(t) = 400e^0.15t. How do you suppose it was done?
   • It was easy to weigh the substance at the start of the experiment. That's where the 400 came from.
   • The question remains: if we know q(t) = 400e^kt, how did the experimenters decide that k was 0.15?

   Apply the natural log function to q(t) = 400e^kt. You will get a linear expression for ln q(t). Explain how you could graph this linear expression to find k.

4. A model for exponential decay (the opposite of exponential growth) might be
   

   q(t) = 650 e-0.07t

   where q(t) is measured in grams and t in hours. You can tell that this model describes decay rather than growth because of the negative coefficient -0.07 in the exponent. Radioactive decay is described by models like this.

   (In radioactive decay, the material at hand doesn't go away: rather, it becomes non-radioactive. A radioactive isotope of carbon might, for instance, be converting itself into a non-radioactive isotope. We'll let the chemists explain all that.)

   1. How much radioactive material was present at the start of the discussion?
   2. After 3 hours, how much material is still radioactive?
   3. What is the half-life of the substance?

5. We have put $2500 in a bank account that pays 5 percent interest at the end of each year.
   1. What will the account balance be at the end of the first year? What is the balance, as a percent of the original $2500? (It will surely be over 100%.)
   2. The years go by, and we have almost forgotten the bank account, although it is still earning 5 percent interest at the end of each year. Suppose that at the beginning of the 10th year the balance in this account is N dollars. Write an expression for its value at the end of the 10th year.
   3. How much is in the account at the end of ten years?
   4. It occurs to us that the bank balance is growing exponentially. Can we rewrite 2500(1.05^t) in the ''usual'' exponential form
      

      2500ekt

      for some number k?
   5. How long does it take for the money in this account to triple?