# Laws of Logarithms

## Laws of Logarithms - How it Works - Video

### Laws of Logarithms ### Base Rule Base Rule:

If the number in the parenthesis is the same as the small number next to the letter g, then the result is 1. A lot of the times, the parenthesis aren't there.

### Log of 1 Rule Log of 1 Rule:

Here we don't have the parenthesis around the big number next to the letter g. If that number is 1, then our answer will be 0. So the log of 1 to base 3 or the log 1 to base 10 or log of 1 base 5 is going to be 0, since any number raised to the power of 0 is 1.

### Product Rule Product Rule:

Here we can separate the logarithm into two logarithms with the two new logarithms keeping the same base as before.

In our example, we log2 (4 * 5) = log2 (4) + log2 (5). Since 4 * 5 = 20, we could have also log2 (10 * 2) = log2 (10) + log2 (2).

Or vice versa, if we have log2 (3) + log2 (5) = log2 (3 * 5) = log2 (15) .

### Quotient Rule Quotient:

Here we can separate the logarithm into two logarithms with the two new logarithms keeping the same base as before.

In our example, we log3 (2 / 7) = log3 (2) - log3 (7).

Or vice versa, if we have log2 (3) - log2 (5) = log2 (3 / 5).

### Power Rule Power Rule:

Here we can separate the logarithm into two logarithms with the two new logarithms keeping the same base as before.

In our example, we log2 (45) = 5 * log2 (4).

Or vice versa, if we have 6 * log2 (3) = log2 (36).

### Reciprocal Rule Reciprocal Rule:

Here we can separate the logarithm into two logarithms with the two new logarithms keeping the same base as before.

In our example, we log4 (1/9) = -log4 (9).

Or vice versa, if we have -log2 (3) = log2 (1/3).

### Change of Base Rule Change of Base Rule:

There are two ways to change the base of logarithms.

The first one log5 (6) = log8 (6) / log8 (5). In this case, the big number is now in the numerator and the small number is the denominator while the base of each new log is 8 or any number.

The second one is log5 (6) = 1/log6 (5). We divided 1 by a new log which we flipped the base and the number.