Converting Between Logarithm Bases

The Formula

Log base b of x equals log base a of x divided by log base a of b, for any base a you like. Choosing a as e gives ln(x) divided by ln(b), which is the version every calculator implements internally.

Why It Works

Because log base a of x and log base a of b are both exponents of the same base, and dividing one exponent by the other converts between bases. It is the same reason you convert between units by dividing by a conversion factor.

Common to Natural

To turn log base 10 into ln, multiply by ln(10), which is about 2.3026. So log base 10 of 100, which is 2, becomes ln(100) of about 4.6052.

Natural to Binary

Multiply by ln(e) divided by ln(2), which is about 1.4427. This is the conversion needed whenever a formula is written in natural logs but the answer is wanted in binary log, common in computing.

Decimal to Binary Log

Multiply by log base 10 of 2, which is about 0.30103. This is the reverse of the previous conversion and is handy when reading tables that mix decimal and binary notation.

Why It Matters in Practice

The change of base identity is the reason information theory uses base 2 while most formulas are written with natural logs. Shannon entropy is conventionally stated in bits, base 2, and the conversion factor is a single constant.