Change of Base Formula Calculator
Enter a logarithm argument, its current base, and the new base you want to convert to. The calculator applies the change of base formula instantly and shows each step of the derivation. Switch between base 10, natural log (base e), base 2, and any custom base. Results update as you type.
Formula
Worked example
Convert log_2(8) to base 10: log_10(8) / log_10(2) = 0.90309 / 0.30103 = 3. Check: 2^3 = 8. Correct.
What is the change of base formula?
The change of base formula converts a logarithm in one base to an equivalent fraction of logarithms in a different base. For a logarithm log_a(x), the formula is: log_a(x) = log_b(x) / log_b(a), where b can be any positive number other than 1. This is useful because most scientific calculators only provide two logarithm keys: log (base 10) and ln (base e). If you need to evaluate log_7(50), for example, you simply compute log(50) / log(7) or ln(50) / ln(7), and both give the same answer.
Why does the formula work?
The derivation starts from the definition of a logarithm. Let p = log_a(x), which means a^p = x. Take the log base b of both sides: log_b(a^p) = log_b(x). Apply the power rule for logarithms (log_b(a^p) = p * log_b(a)): p * log_b(a) = log_b(x). Solve for p: p = log_b(x) / log_b(a). Since p was defined as log_a(x), we arrive at log_a(x) = log_b(x) / log_b(a). The formula holds for any base b, so the choice of b is only a matter of what your calculator supports.
How to use this calculator
Enter the argument (the number x inside the logarithm), the original base (a), and select the new base (b). Base 10 and base e are the most practical choices because they match the log and ln keys on a calculator. For custom bases, switch the new base selector to "Custom base" and enter any positive number other than 1. The calculator shows the numerator log_b(x), the denominator log_b(a), their quotient (the result), and a verification step that raises a to the result to confirm it equals x.
Special cases and logarithm properties
Several special values follow directly from the definition. log_a(1) is always 0 because a^0 = 1 for any base. log_a(a) is always 1 because a^1 = a. If x equals a power of the base (such as log_2(8) = 3), the result is a whole number. The product rule says log_a(xy) = log_a(x) + log_a(y). The quotient rule says log_a(x/y) = log_a(x) - log_a(y). The power rule says log_a(x^n) = n * log_a(x). All of these hold regardless of the base, which is why the change of base formula is so powerful: once you can evaluate a logarithm in one base, you can evaluate it in any base.
Common logarithm identities and special values
| Expression | Value | Reason |
|---|---|---|
| log_a(1) | 0 | a^0 = 1 for any base a |
| log_a(a) | 1 | a^1 = a for any base a |
| log_10(10) | 1 | Base 10 raised to 1 equals 10 |
| log_10(100) | 2 | 10^2 = 100 |
| log_10(1000) | 3 | 10^3 = 1000 |
| ln(e) | 1 | e^1 = e |
| ln(e^2) | 2 | e^2 = e^2 |
| log_2(2) | 1 | 2^1 = 2 |
| log_2(8) | 3 | 2^3 = 8 |
| log_2(1024) | 10 | 2^10 = 1024 |
These values follow directly from the definition of a logarithm and hold for any valid base.
Frequently asked questions
What is the change of base formula?
The change of base formula states that log_a(x) = log_b(x) / log_b(a) for any valid positive base b (not equal to 1). It lets you rewrite any logarithm as a ratio of two logarithms in a base your calculator supports, typically base 10 or base e.
Why do we need the change of base formula?
Most scientific calculators and spreadsheet functions only have log (base 10) and ln (base e) built in. If you need log_7(50) or log_3(100), you cannot enter those directly. The change of base formula rewrites them as log(50)/log(7) and log(100)/log(3), which any calculator can handle.
Does it matter which new base I choose?
No, the result is the same regardless of which new base you pick. log_a(x) = log_10(x)/log_10(a) = ln(x)/ln(a) = log_2(x)/log_2(a). The numerator and denominator both change, but their ratio stays constant. In practice, base 10 or base e is the most convenient because those are built into calculators.
How do I convert log base 2 to base 10?
Use log_2(x) = log(x) / log(2). For example, log_2(64) = log(64) / log(2) = 1.80618 / 0.30103 = 6. You can verify this because 2^6 = 64.
What values are not allowed as a base or argument?
The argument x must be positive (greater than zero). Logarithms of zero or negative numbers are not real numbers. The base must be positive and not equal to 1. A base of 1 gives log_1(x), which has no solution because 1 raised to any power is still 1. The new base b must satisfy the same conditions.
How do I verify my logarithm answer?
Raise the original base to the result. If log_a(x) = r, then a^r should equal x. For example, if log_2(8) = 3, then 2^3 = 8. This calculator shows the verification step automatically.