### 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

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 (ab
^{2}) + log (a^{2}b) = log (a^{3}b^{3})

### Adding two logarithms with different bases:

log_{m}2 + log_{m}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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