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
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)
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.