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Addition of Logarithms

Law of Addition of Logarithms

`log(a) + log(b) = log(ab)`
If the log has a base 'm', then the above law can be written as
`log_m(a) + log_m(b) = log_m(ab)`
The above law can be used to add two logs with same bases. If we write it in reverse order, it becomes:
`log_m(ab) = log_ma + log_mb`
... which can be used to split a logarithm into a sum of two logs (with same bases). (If you don't know what the word 'base' here, check out this post.)
In the above equations,
  • `a` and `b` can be any numbers/expressions
  • `m` is the base of the logarithms
Note that the base of all logarithms above is the same. The above law of addition of logarithms is only applicable when the logarithms have the same bases. All of the above logarithms have base 'm', which can be any positive real number. Thus, two logarithms when added, result in a logarithm of their product.

For example,
log 2 + log 3 = log (2 * 3)
So,
log 2 + log 3 = log 6

Some more examples:

  •     log x + log y = log (xy)
  •     log 2a + log 5b = log (10ab)
  •     log (ab2) + log (a2b) = log (a3b3)
Note that in the above examples, the bases of each logarithm are assumed to be same. In case of logarithms with different bases, first convert their bases to the same base, and then simplify the logarithms.

Adding two logarithms with different bases:

logm 2 + logm 3 = ?

The base of log 2 is 'm' and that of log 3 is 'n'. Thus, the above logarithms have different bases.
How do you add these logarithms?

1. Make their bases same by applying the base changing formula. Thus,


2. Now you have a rational expression in logarithms. Simplify it as any other rational expression:

, which is the answer.

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