 ## 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

` bx = 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 = logby   <--->  bx = 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`
• ` 251/2 = 5      log255 = 1/2`
• ` 8-2/3 = 1/4      log81/4 = -2/3`
• ` 21/2 =  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.