How to Calculate Logarithms

What a Logarithm Means

Log base b of x equals n exactly when b raised to the power of n equals x. Finding log base 2 of 8 is therefore the same question as asking which power of 2 gives 8, and the answer is 3 because 2 times 2 times 2 is 8.

Spot the Pattern First

Before calculating, check whether your value is a whole power of your base. Powers of 10 give whole number logs immediately, and powers of 2 are easy to spot from their binary form. Recognising the pattern turns a calculation into a single glance.

The Change of Base Formula

Any logarithm can be rewritten as ln(x) divided by ln(b). This is what makes a single calculator button able to serve every base, and it means you never need a special formula for common log, natural log or binary log, only the natural log.

A Worked Example

Find log base 3 of 81. The natural log of 81 is about 4.3944 and the natural log of 3 is about 1.0986, so the quotient is exactly 4, confirmed by 3 to the power of 4 being 81.

Which Inputs Are Valid

Only positive values have a real logarithm, and only positive bases other than 1 are allowed. Zero gives an answer of minus infinity in the limit, negative values have no real log, and base 1 can never produce anything but 1.

Negative Answers Are Normal

Any value between 0 and 1 gives a negative logarithm, because raising a base greater than 1 to a negative power gives a fraction. Log base 10 of 0.01 is minus 2, which is a correct and useful answer rather than a problem.