Laws of Logarithms

Logarithms follow three important rules. The log of a product is the sum of the logs: log 6 = log 2 + log 3. The log of a quotient is the difference of the logs. And an exponent can be moved in front of the log: log 2⁵ = 5 · log 2. Thanks to these rules you can work out log₆ 4 + log₆ 9 without a calculator — it is log₆ 36 = 2.

What does this generator do?

On the easy level you add and subtract logarithms with the same base, for example log₆ 4 + log₆ 9. On the medium level the power rule comes in: 2 · log₃ 6 − log₃ 4. On the hard level you expand the logarithm of a fraction, condense an expression into a single logarithm and evaluate the product log₂ 3 · log₃ 8.

How to use it

1) Move the number in front of the logarithm into the exponent: 2 · log₃ 6 = log₃ 36. 2) Turn a sum of logs into the log of a product, and a difference into the log of a quotient. 3) Evaluate the result from the definition: log₃ 9 = 2. 4) For different bases use log_a b = log b / log a.