# 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 log_{2} (4 * 5) = log_{2} (4) + log_{2} (5). Since 4 * 5 = 20, we could have also log_{2} (10 * 2) = log_{2} (10) + log_{2} (2).

Or vice versa, if we have log_{2} (3) + log_{2} (5) = log_{2} (3 * 5) = log_{2} (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 log_{3} (2 / 7) = log_{3} (2) - log_{3} (7).

Or vice versa, if we have log_{2} (3) - log_{2} (5) = log_{2} (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 log_{2} (4^{5}) = 5 * log_{2} (4).

Or vice versa, if we have 6 * log_{2} (3) = log_{2} (3^{6}).

### Reciprocal Rule

Reciprocal Rule:

In our example, we log_{4} (1/9) = -log_{4} (9).

Or vice versa, if we have -log_{2} (3) = log_{2} (1/3).

### Change of Base Rule

Change of Base Rule:

There are two ways to change the base of logarithms.

The first one log_{5} (6) = log_{8} (6) / log_{8} (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 log_{5} (6) = 1/log_{6} (5). We divided 1 by a new log which we flipped the base and the number.