Change of Base Calculator

Convert logarithms between any bases using the change of base formula. See step-by-step solutions.

Enter Logarithm Values

The argument of the logarithm (must be positive)
The base of the logarithm you want to compute
Convert to this base (common choices: 10, e, 2)

Result

log_10(100)
--
exact value
log_b(x) result --
log_a(x) --
log_a(b) --
ln(x) (natural log) --
log_10(x) (common log) --
log_2(x) (binary log) --

Step-by-Step Solution

log_b(x) = log_a(x) / log_a(b)

What Is the Change of Base Formula?

The change of base formula allows you to compute a logarithm with any base using logarithms of a different base. This is particularly useful because most calculators only have buttons for common logarithms (base 10) and natural logarithms (base e). The formula states:

log_b(x) = log_a(x) / log_a(b)

Where b is the original base, x is the argument, and a is any new base you choose. The most common choices for a are 10 (common logarithm), e (natural logarithm), or 2 (binary logarithm).

Change of Base Formulas

General Formula

Convert log base b to any other base a.

log_b(x) = log_a(x) / log_a(b)

Using Common Log (base 10)

Use the log button on your calculator.

log_b(x) = log(x) / log(b)

Using Natural Log (base e)

Use the ln button on your calculator.

log_b(x) = ln(x) / ln(b)

Reciprocal Property

The logarithm of a base in terms of another.

log_b(a) = 1 / log_a(b)

How to Use the Change of Base Formula

  1. Identify the logarithm: You want to compute log_b(x), where b is the base and x is the argument.
  2. Choose a new base: Pick a base that your calculator supports. Base 10 (log) and base e (ln) are the most common.
  3. Compute the numerator: Calculate log_a(x), the logarithm of x in the new base.
  4. Compute the denominator: Calculate log_a(b), the logarithm of the original base in the new base.
  5. Divide: The result log_a(x) / log_a(b) gives you log_b(x).

Example: Calculate log_5(125)

Using common logarithms (base 10): log_5(125) = log(125) / log(5) = 2.09691 / 0.69897 = 3. This makes sense because 5^3 = 125, so log_5(125) = 3.

Why the Change of Base Formula Works

The formula derives from the definition of logarithms. If log_b(x) = y, then b^y = x. Taking log base a of both sides: log_a(b^y) = log_a(x). By the power rule of logarithms: y * log_a(b) = log_a(x). Solving for y: y = log_a(x) / log_a(b). Since y = log_b(x), we have the change of base formula.

Common Logarithm Bases

  • Base 10 (Common Log): Written as log(x) or log_10(x). Used in scientific notation, decibels, pH scale, and Richter scale.
  • Base e (Natural Log): Written as ln(x). Used extensively in calculus, exponential growth/decay, and continuous compound interest.
  • Base 2 (Binary Log): Written as log_2(x) or lb(x). Essential in computer science for analyzing algorithm complexity and information theory.

Applications

  • Computer Science: Analyzing algorithm time complexity (e.g., binary search is O(log_2 n)).
  • Information Theory: Computing entropy and information content using base-2 logarithms (bits).
  • Chemistry: pH calculations use base-10 logarithms of hydrogen ion concentration.
  • Finance: Calculating doubling time and compound interest using natural logarithms.
  • Acoustics: Decibel calculations use base-10 logarithms of power ratios.