changeofbase

Change of base is the process of converting numbers or expressions from one numeral base to another. In mathematics and computer science, this concept is used for both logarithms and numeral systems.

In logarithms, the change of base formula allows evaluating log_b(x) using any other base k. The formula is log_b(x) = log_k(x) / log_k(b). This is useful because calculators typically provide logs in a standard base, such as 10 or e. For example, log_3(12) can be computed as ln(12)/ln(3) or log_10(12)/log_10(3). The formula requires x > 0 and bases b > 0, b ≠ 1, k > 0, k ≠ 1.

In numeral systems, change of base refers to converting a representation from base a to base b. For the integer part, divide by b repeatedly and record remainders; the remainders read in reverse give the base-b digits. For the fractional part, repeatedly multiply by b and take the integer part as the next digit. This method converts numbers such as binary 1011_2 to decimal 11, or decimal 255_10 to hexadecimal FF_16. Bases typically extend up to 36 using digits 0–9 and letters A–Z for digits above 9.

Key properties include that log_b(x) is defined for x > 0 and bases b > 0, b ≠ 1, with log_b(x) = log_k(x)/log_k(b). A number’s base representation is not unique in form but represents the same value; practical conversions must manage rounding for fractional parts.

Applications span mathematics, computing, and data encoding, highlighting that a number’s representation depends on the chosen base while its value stays the same under a consistent base change.