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 log_a1 = 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 log_aa = 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,   log_aM.N = log_aM + log_aN.

Proof

If M.N is the product and if a^x = M and a^y = N, then M.N = a^x . a^y.

By the law of indices  a^x. a^y = a^x+y. Therefore,

a^x+y  = M.N

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

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

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

log_a (M.N) = log_a M + log_a 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, log_a (M/N) = log_a M - log_a N.

Proof

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

      log_a (M/N) = x - y .......(1)

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

      log_a (M/N) = log_a M - log_a 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, log_a (M^P) = p.log_aM.

Proof

If M = a^x, then log_a M = x. Now suppose that M is raised to the power of n, that is Mⁿ. Since M = a^x, Mⁿ = a^nx. 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 = a^nx, if expressed in terms of logarithms will be

      log_a(Mⁿ) = nx      ...........(1)

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

      log_a (Mⁿ) = n . log_a M

Similarly if n = 1/r, we have

      log_a (M^1/r) = (1/r) . log_a 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.