Change of Base Formula Calculator

This calculator applies the change of base formula to rewrite any logarithm in a new base, letting you evaluate it with a standard calculator or a table. Enter the argument (the number you take the log of), the original base, and the new base, and you get the logarithm value instantly. The formula works for any positive bases (except 1) and any positive argument.

Formula

For any positive numbers x, b, and c, where b ≠ 1 and c ≠ 1, the change of base formula is:

log_b(x) = log_c(x) / log_c(b)

Equivalently, using natural logarithms: log_b(x) = ln(x) / ln(b). The calculator above uses this second form with the new base c you choose, computing log_c(x) = ln(x) / ln(c). All variables are unitless numbers.

Worked Example

Using the default values: x = 100, b = 10, c = 2. We want to find log₂(100).

1. Compute ln(100) ≈ 4.605170
2. Compute ln(2) ≈ 0.693147
3. Divide: 4.605170 / 0.693147 ≈ 6.643856

So log₂(100) ≈ 6.6439. The calculator shows 6.6439 as the headline answer.

Why the Change of Base Formula Matters

Most scientific calculators only have buttons for log base 10 (log) and natural log (ln). Before the formula, evaluating a logarithm with an arbitrary base like 5 or 7 required a separate table or a lot of manual work. The change of base formula lets you convert any logarithm into a ratio of logs you can actually compute. It also appears in algebra, calculus, and computer science—for example, when comparing algorithmic complexities (log₂ vs log�₀) or solving exponential equations.

How to Choose the New Base

• Use ln (base e) when working with calculus, natural growth, or if your calculator has an ln key.
• Use log�₀ when dealing with pH, decibels, or other base-10 applications.
• Use base 2 for binary systems, information theory, or algorithms.
• Any base works mathematically; the choice only affects the numbers you type.

Common Mistakes and How to Avoid Them

• Forgetting that bases must be positive and not equal to 1. A base of 1 is invalid because 1 raised to any power is 1, so log�(x) is undefined.
• Swapping numerator and denominator. The formula is log_c(x) / log_c(b), not log_c(b) / log_c(x). Always put the argument in the numerator.
• Using the original base in the denominator. The denominator uses the new base, not the original base.
• Rounding too early. Keep several decimal places in intermediate steps to get an accurate final answer.
• Confusing log and ln. On many calculators, the log key is base 10 and ln is base e. Make sure you know which one you’re using.

Limitations of the Formula

The change of base formula only works for positive arguments and positive bases (excluding 1). It does not apply to complex numbers or to logarithms with a base of 0 or negative numbers. Also, the formula gives an exact value only when you keep enough precision; a calculator’s floating-point arithmetic may introduce tiny rounding errors for very large or very small inputs. For most practical purposes, four to six decimal places are sufficient.

Frequently Asked Questions

Can I use any base as the new base?

Yes, any positive number except 1 works. The result is the same regardless of which base you choose, because the formula adjusts the ratio accordingly. For example, log₂(8) = 3 whether you use ln or log�₀.

Why does the change of base formula work?

It follows from the definition of logarithms and the property that log_c(b) * log_b(x) = log_c(x). Dividing both sides by log_c(b) gives the formula. In essence, you’re scaling the logarithm by a constant factor to convert from one base to another.

What if my argument is less than 1?

The formula still works. For example, log₂(0.5) = -1, because 2�¹ = 0.5. The calculator will give a negative result, which is correct. Just ensure the argument is positive; logarithms of zero or negative numbers are undefined in real numbers.

How do I use this formula on a scientific calculator?

Suppose you want log₅(20). Press ln(20) ÷ ln(5) or log(20) ÷ log(5) and press equals. The result is approximately 1.861. That’s exactly what the change of base formula does.

Is the change of base formula the same as the quotient rule for logarithms?

No. The quotient rule says log_b(x/y) = log_b(x) − log_b(y). The change of base formula is a different property that relates logarithms of different bases. They are both useful but serve different purposes.

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in x | Argument (x) | - | 100 | 0
in b | Original base (b) | - | 10 | 0
in c | New base (c) | - | 2 | 0
out result | log_c(x) | - | ln(x) / ln(c) | 4