The logarithm of a number is the power to which you must raise the base in order to produce this number. The logarithm with base 10 is called the common logarithm and denoted as “log.” For example, log(1,000) is 3, as 10 raised in the power of 3 produces 1,000. Every scientific calculator has a built-in function to calculator log of any number (typically the button “log”). But you rarely see a calculator which performs a log_{2} function, which is logarithm with base 2, directly. As an example, calculate log_{2} of the number “12” i.e. log_{2}(12).

To calculate the base 2 logarithm of a number (y), divide the common log of y by the common log of 2.

## Set up the Expression

Express log_{2}(y) of any number y via log(y). According to the logarithm definition y=2^{(log2(y))}. Take log of both sides of the equation to get log(y)=log(2^{(log2(y)}) = log(2) × log_{2}(y). Then divide both sides by log(2) and rearrange to get log_{2}(y)=log(y)÷log(2).

## Calculate Log(2)

Calculate log(2) with a calculator. Enter “2” and press the “log” button. log(2)=0.30103. Write down this constant as it will be used in all calculations of log_{2}.

## Calculate Log(y)

Calculate log(y). Enter a number and press the “log” button. In our example, log(12)= 1.07918.

## Calculate Log2(y)

Divide the result from the last step by the constant log(2) obtained above to get log_{2}(y). In our example, it would be log_{2}(12)=log(12)÷log(2)=1.07918÷0.30103 = 3.584958.

#### References

- “Algebra II (Cliffs Quick Review)”, E. Kohn and D.A. Herzog, May 29, 2001.
- Richland Community College: Properties of Logarithms