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.
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_{b}y <---> 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.
10^{2} = 100 log_{10}100 = 2
10^{-2} = 0.01 log_{10}0.01 = -2
10^{0} = 1 log_{10}1 = 0
2^{3} = 8 log_{2}8 = 3
3^{2} = 9 log_{3}9 = 2
25^{1/2} = 5 log_{25}5 = 1/2
8^{-2/3} = 1/4 log_{8}1/4 = -2/3
2^{1/2} = 1.4142135623... log_{2}1.414.. = 1/2
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:
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]