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.