Rules of logarithms, Mathematics

Rule 1

The logarithm of 1 to any base is 0.

Proof

We know that any number raised to zero equals 1. That is, a0 = 1, where "a" takes any value. Therefore, the logarithm of 1 to the base a is zero. Mathematically, we express this as loga1 = 0.

Example 

What is the value of log101.

Needless to say this would be zero.

Rule 2

The logarithm of a number where the number is the same as the base is 1.

Proof

We know that any number raised to the power of 1 is itself. That is a1 = a. Therefore, the logarithm of  a to the base a is equal to 1.

Mathematically, we express this as logaa = 1.

Example 

What is the value of log1313?

By applying the above rule, the value of log1313 is 1.

Rule 3

The logarithm of a product to base a is equal to sum of the logarithms of the individual numbers which constitute the product to the same base a. That is,   logaM.N = logaM + logaN.

Proof

If M.N is the product and if ax = M and ay = N, then M.N = ax . ay.

By the law of indices  ax. ay = ax+y. Therefore,

ax+y  = M.N

Then the logarithm of M.N to base a is equal to x + y. Mathematically, it will be

loga M.N = x + y                                                      ......(1)

Now, if we express ax = M and ay = N, in terms of logarithms they will be               loga M = x and loga N = y. Substituting the values of x and y in 1, we have

loga (M.N) = loga M + loga N

Example 

What is the value of log333?

We know that 33 can be expressed as the product of 3 and 11. That is,    log3 33 = log3 (3 x 11). Applying the above rule this can be expressed as log3 3 + log3 11. Since log33 is 1, we rewrite it as log3 33 = 1 + log3 11.

Rule 4

The logarithm of a fraction to the base a will be equal to the difference of the logarithm of the numerator to the base a and the logarithm of the denominator to base a. That is, loga (M/N) = loga M - loga N.

Proof

Let ax = M and ay = N. Then M/N = ax/ay. By the law of indices, this will equal to ax-y. The logarithm of M/N to base 'a' will, therefore, be x - y. Mathematically this is expressed as

      loga (M/N) = x - y .......(1)

If we express ax = M and ay = N in terms of logarithms, they will be loga M = x and loga N = y. Substituting the values of x and y in (1), we have

      loga (M/N) = loga M - loga N

Example 

What is the value of log2 (32/4).

By applying the above rule, this can be written as log2 32 - log2 4. This can be further solved. But we look at it only after learning the next rule.

Rule 5

The logarithm of a number raised to any power, integral or fractional, is equal to product of that number and the logarithm of the number raised to base a. That is, loga (MP) = p.logaM.

Proof

If M = ax, then loga M = x. Now suppose that M is raised to the power of n, that is Mn. Since M = ax, Mn = anx. This is in accordance with the priniciple that if we perform any operation on an equation it should be performed on both the sides of the equation in order to keep the equation symbol valid.

Mn = anx, if expressed in terms of logarithms will be

      loga(Mn) = nx      ...........(1)

On substituting loga M = x in (1), we have

      loga (Mn) = n . loga M

Similarly if n = 1/r, we have

      loga (M1/r) = (1/r) . loga M

Now we take up the example discussed under Rule 4 and look at how it is further simplified. Before we go on to the next step, let us express log2 32 and log2 4 as log2 25 and log2 22. By rule 5, these are expressed as 5.log22 and 2.log2 2. And since log2 2 is one, 5.log22 and 2.log22 reduce to 5.1 = 5 and 2.1 = 2. Therefore, log2 32 - log24 when simplified gives

  log2(25) - log2(22)

   =   5.log22 - 2.log22

   =   5.1 - 2.1

   =   5 - 2 = 3.

We obtain the same value even by simplifying the term on the left hand side. We know that 32/4 = 8. That is, log28 can be expressed as 23. On application of rule 5, this will be 3.log2 2. Again this gives us 3.1 = 3.

Generally, logarithms are expressed to base 10 and base 'e'. While the logarithms expressed to base 10 are referred to as common logarithms, those expressed to base 'e' are referred to as Napier or Natural logarithms. The value of 'e' is approximately 2.718. In practise common logarithms are expressed as 'log' while natural logarithms are expressed as 'ln'. We want to emphasize that generally the base is not stated and by looking at the manner it is expressed we ought to decide whether it is a common or natural logarithm.

Posted Date: 9/13/2012 5:40:39 AM | Location : United States







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