The Question a Logarithm Asks
Log base 10 of 1000 asks: ten to what power gives 1000. The answer is 3. That is all a logarithm is, an exponent question asked in reverse, and once you read it that way most of the mystery goes.
So log of 100 is 2, log of 10 is 1, and log of 1 is 0, because anything to the power of zero is one.
log and ln
On a calculator, log usually means base ten and ln means base e, where e is roughly 2.718. Base ten suits anything measured in orders of magnitude; base e appears wherever something grows or decays continuously, which is most of physics, biology and compound interest.
The button labels differ by manufacturer, which is worth checking before an exam rather than during one.
The Three Laws
The log of a product is the sum of the logs. The log of a quotient is the difference. The log of a power brings the exponent out to the front as a multiplier.
That third law is the one that earns its keep, because it turns an unknown sitting in an exponent into an ordinary coefficient you can divide by. It is how equations like 2 to the x equals 50 get solved at all.
Change of Base
Calculators generally offer only base ten and base e, which is inconvenient when a question asks for base 2. Divide: log base 2 of 32 equals log 32 divided by log 2, which gives 5. Either available base works, provided you use the same one top and bottom.
Why Log Scales Are Everywhere
Earthquake magnitude, sound in decibels and acidity in pH are all logarithmic, because the underlying quantities span too many orders of magnitude to plot honestly on a linear axis.
It also explains why a magnitude 6 earthquake is not slightly worse than a 5 but roughly thirty times more energetic. Each step up the scale is a multiplication, not an addition.
