Change of Base Calculator

The change of base formula states that log_a(x) = log_b(x) / log_b(a), where b can be any valid base (commonly 10 or e). This allows you to compute a logarithm in any base using only a standard calculator that supports base-10 or natural logarithms.

Most calculators and programming languages only support base-10 (log) and base-e (ln) logarithms. The change of base formula lets you convert a logarithm from an inconvenient base—like base 2 or base 27—into one your calculator can handle directly.

Identify your argument x, original base a, and desired new base b. Then compute log_b(x) and log_b(a) separately, and divide: log_a(x) = log_b(x) ÷ log_b(a). For example, log_27(9) = log(9) / log(27) ≈ 0.9542 / 1.4314 ≈ 0.6667.

Use the formula: log_2(x) = log₁₀(x) / log₁₀(2). Since log₁₀(2) ≈ 0.30103, you simply divide the base-10 logarithm of x by 0.30103. For example, log_2(8) = log(8) / log(2) = 0.9031 / 0.3010 = 3.

You can convert using: log₁₀(x) = ln(x) / ln(10). Since ln(10) ≈ 2.302585, divide the natural log of x by 2.302585. Conversely, ln(x) = log₁₀(x) × ln(10) ≈ log₁₀(x) × 2.302585.

No. Log base 2 (binary logarithm) uses 2 as its base, while the natural logarithm uses Euler's number e ≈ 2.71828. They are related by the change of base formula: log_2(x) = ln(x) / ln(2) ≈ ln(x) / 0.6931.

The logarithm of 0 is undefined. As the argument x approaches 0 from the positive side, log_a(x) approaches negative infinity. You must always use a positive value for x when computing any logarithm.

Yes, as long as the new base b is positive and not equal to 1, and the original base a is also positive and not equal to 1. The most common choices for b are 10 and e, since calculators natively support those, but any valid base works mathematically.