Hi .....Plz explain Logarithms and their properties...

Logarithms, or "logs", are a way of expressing one number in terms of a "base" number that is raised to some power. Common logs are done with base ten, but some logs ("natural" logs) are done with the constant "e" (2.718 281 828) as their base. The log of any number is the power to which the base must be raised to give that number.

We are most familiar with base 10 since any number greater than zero can be expressed as 10^x. For example, log(10) is 1 (because 10¹ =10), log(100) is 2 (because 10² = 100), and log(2) is 0.3 (because 10^0.3 ≈ 2). Logs can easily be found for either base on your calculator. Usually there are two different buttons, one saying "log", which is base ten, and one saying "ln", which is a natural log, base e. It is always assumed, unless otherwise stated, that "log" means log₁₀.

The opposite of a log is the antilog, which means to raise the base to that number. Antilogs "undo" logarithms. Observe the following example:

Logs are read aloud as "log", "natural log", "ln", or "log base whatever". To read log₃4, you would simply say "log, base three, of four".

Logs are commonly used in chemistry. The most prominent example is the pH scale. The pH of a solution is the -log([H⁺]), where square brackets mean concentration.

There are two major kinds of equations that you will have to solve using logs. In one kind, you will know the log of a number and have to find the number by taking anti-logs, which means raising the base to a power. The other kind gives you the variable in the exponent, and you have to take logs to isolate it. Solving these kinds of problems depends on knowing another property of logs: if the log of a number with an exponent is taken, then the log of that number is multiplied by whatever was in the exponent.

The Four Basic Properties of Logs

1.      logb(xy) = logbx + logby.
2.      logb(x/y) = logbx - logby.
3.      logb(xn) = n logbx.         
4.      logbx = logax / logab.     

These four basic properties all follow directly from the fact that logs are exponents. In words, the first three can be remembered as: The log of a product is equal to the sum of the logs of the factors. The log of a quotient is equal to the difference between the logs of the numerator and demoninator. The log of a power is equal to the power times the log of the base.

Additional properties, some obvious, some not so obvious are listed below for reference. Number 6 is called the reciprocal property.

   1. logb1 = 0.         
   2. logbb = 1.         
   3. logbb2 = 2.        
   4. logbbx = x.        
   5. blogbx = x.         
   6. logab = 1/logba.