What on Earth is a Logarithm?

Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

What on Earth is a Logarithm?

Interestingly, after I had this guide up for a while, this turned out to be the question I was asked most frequently, usually in terms that included phrases like "Greek to me", "beats me", or, as above, "what on earth"...

To understand what a logarithm is you first have to understand what a power is. Follow that link first if you don't!

OK, you do know what a power is. So it makes sense to you to write something like

 b^x = y.      (*)

In the preceding equation, the x should look like a superscript of b. If it does not you have an underpowered browser. Replace it with one like Netscape.

After these preliminaries, we can now get into the meat of the matter. The equation (*) is the key to everything. The number b is the base, the number x the exponent, and the expression that equals y is a power. If we think of x as the independent variable and y as the dependent variable then (*) defines an exponential function.

In the equation (*) we can now pretend that two of the variables are given, and solve for the third. If the base and the exponent are given we compute a power, if the the exponent and the power are given we compute a root (or radical ), and, if the power and the base are given, we compute a logarithm.

In other words, The logarithm of a number y with respect to a base b is the exponent to which we have to raise b to obtain y.

We can write this definition as

x = log_by   <--->  b^x = y

and we say that x is the logarithm of y with base b if and only if b to the power x equals y.

Let's illustrate this definition with a few examples. If you have difficulties with any of these powers go back to my page on powers.

 10² = 100      log₁₀100 = 2

 10^-2 = 0.01      log₁₀0.01 = -2

 10⁰ = 1      log₁₀1 = 0

 2³ = 8      log₂8 = 3

 3² = 9      log₃9 = 2

 25^1/2 = 5      log₂₅5 = 1/2

 8^-2/3 = 1/4      log₈1/4 = -2/3

 2^1/2 =  1.4142135623...      log₂1.414.. = 1/2

Special Bases

Logarithms with respect to the base b=10 are called common logarithms, and logarithms with respect to the base e=2.71828... are called natural logarithms.

More Information

You should find extensive information on logarithms in any textbook on College Algebra. To check your understanding and guide your further study figure out answers to the following questions:

Why are logarithms important?
Why are exponential functions important?
How do you convert a logarithm with respect to one base to a logarithm with respect to another base?
Why does the base have to be positive?
Why is the power always positive?
What is it that makes natural logarithms natural?

A Logarithm Calculator

Click on this applet

However, your browser does not support Java. If it did you would not see this message! Get a java compatible browser such as Netscape, of a sufficiently advanced version.

to bring up a Logarithm Calculator that lets you pick two of the numbers in (*) and computes the third. It's pretty straightforward to use, but here is documentation.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[27-Jun-1997]