When teaching mathematics I often find that students struggle with the concept of logarithms. You can think of the logarithm as the inverse operation of exponents. Let's start with some basic variables; x, b, and y are commonly used when dealing with this subject. We have some number "x", a base "b" and an exponent "y" such that x = b^{y}. By definition, the logarithm of some number "x" of a given base, "b" is the power to which the base must be raised in order to produce that number ("x").

So if x = b^{y}, then log_{b}(x) = y.

Consider a practical example. If you are asked to find log_{10}(100), how would you tackle this? Let's start by identifying what we know and what we need to find using the same variables we started with above. In this example, we have a base b=10 and some number x=100. Knowing what we now know about logarithms, that is, if log_{b}(x) = y, then x = b^{y}, then we simply plug in the numbers we do not know and see if we can find the numbers we do not. So if b = 10 and x = 100, then from the definition, 100 = 10^{y}. This is a fairly simple problem to solve because we know that 10^{2} = 100, so y = 2. So, going back to the original question, we now know that log_{10}(100) = 2.

When dealing with logarithms, a base of 10 is quite common. For this reason, rather than write log_{10}(x), we simply write log(x). Only in cases where we use a base other than 10 do we actually write the base. The number e is also a common base, so rather than write log_{e}(x), we use the special ln(x) to designate the natural logarithm.

Special Properties of Logarithms

With some of the basics out of the way, we can move on to the properties of logarithms:

1. The logarithm of the product of two numbers is equivalent to the sum of the logarithms of each number: log (xy) = log (x) + log (y)

2. The logarithm of a quotient is equivalent to the logarithm of the numerator minus the logarithm of the denominator: log (^{x}/_{y}) = log (x) - log (y)

3. The logarithm of a power of x is equivalent to that exponent times the logarithm of x: log (x^{n}) = n log (x)

Changing Base

Because your calculator only does logarithm calculations using base 10 or base e, it is often necessary to "change base" when doing logarithm mathematics. For example, you can quite easily get a result from your calculator for log (92). But what if you are asked to find log_{7}(92)? For this you will need the change of base formula: log_{b} (x) = log (x) / log (2).

This should help get you rolling with basic logarithm problem solving. Stay tuned for more algebra and mathematics help right here on InfoBarrel.