Logarithm Laws and Properties
The laws of logarithms are the identity rules of the topic. They let you break up problems that look impossible in their current form and are the reason logarithms work for solving exponential equations at all.
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Try the Logarithm CalculatorLaw One: Product Becomes Sum
log base b of x times y equals log base b of x plus log base b of y. This turns a multiplication problem into an addition, which is why logarithms appear so often in algorithmic complexity work.
Law Two: Quotient Becomes Difference
log base b of x divided by y equals log base b of x minus log base b of y. It is the quotient form of the product law and is used constantly in equilibrium and growth calculations.
Law Three: Power Comes Out Front
log base b of x to the power p equals p multiplied by log base b of x. This is the law that makes log base 2 of 8 to the power of 3 equal 9, since the log of 8 is 3 and 3 times 3 is 9.
The Two Special Cases
Log base b of 1 is always 0, in every base without exception. Log base b of b is always 1. Both follow from the zero and first power laws and make excellent checks on any answer.
The Identity and Inverse Laws
Log base b of b to the power of x is x, and b to the power of log base b of x is x. These are the inverse pairs that connect logarithms and exponentials, and they are what make an antilog the exact reverse of a log.
Why Logs of Negatives Fail
Every logarithm law requires a positive argument. Since any real power of a positive base is positive, there is no way to produce a negative number as an output, so a real logarithm of a negative number cannot exist. Algebra that appears to give one has made a mistake somewhere.
Worked Example
log base 2 of 8 times 4 equals log base 2 of 8 plus log base 2 of 4, which is 3 plus 2, giving 5. Check: 8 times 4 is 32, and 2 to the power of 5 is 32.
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