Change of Base Calculator

Enter x (the argument), a (the original base), and b (the new base) to convert any logarithm using the change of base formula. The Change of Base Calculator computes log_a(x) by rewriting it as log_b(x) ÷ log_b(a), giving you the result in your chosen base alongside the intermediate values. Also try the find Final Value x(t) with Exponential Decay Calculator.

The value whose logarithm you want to find. Must be greater than 0.

The base of the original logarithm. Must be > 0 and ≠ 1.

The base to convert into (e.g. 10 for common log, 2.718 for natural log). Must be > 0 and ≠ 1.

Results

log_a(x) Result

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log_b(x) — Numerator

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log_b(a) — Denominator

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log₁₀(x) — Common Log of x

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ln(x) — Natural Log of x

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What is the change of base formula?

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. See also our calculate Antilog Result (x = b^y), Base Used & Log Value Used (y) — Antilog.

Why do we need the change of base formula?

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.

How do I change the base of a logarithm step by step?

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.

How do you change log base 2 to base 10?

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 might also find our Logarithm Calculator (log) useful.

How do you change log base 10 to base e (natural log)?

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.

Is log base 2 the same as the natural log?

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.

What is the logarithm of 0?

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.

Can the change of base formula be used for any base?

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.