Logarithm Calculator

What the Logarithm Calculator Does and Who It's For

This logarithm calculator finds the logarithm of a positive number for any base you choose. Enter the value (x) and the base (b), and it returns log_b(x) — the exponent to which the base must be raised to produce your number.

It's useful for students working through algebra and precalculus, engineers and scientists dealing with decibels, pH, or signal scales, and anyone who needs a quick log without a scientific calculator. You can compute common logs (base 10), binary logs (base 2), or the natural logarithm, which uses the constant e (about 2.71828).

How Logarithms Work: The Formula

A logarithm answers the question: "What power do I raise the base to in order to get this number?" Formally:

log_b(x) = y means b^y = x

So log_b(x) is the inverse of exponentiation. The base b must be positive and not equal to 1, and the value x must be positive. Logarithms of zero or negative numbers are undefined in the real numbers.

Calculators usually store only base-10 (log) and base-e (ln) functions, so to find any other base they apply the change-of-base formula:

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

Worked Example with Real Numbers

Suppose you want log_2(32) — the power of 2 that gives 32. Since 2^5 = 32, the answer is exactly 5.

Now try a value that isn't a clean power, such as log_2(50). Using change of base: log_2(50) = ln(50) / ln(2) = 3.91202 / 0.69315 = 5.6439. You can check it: 2^5.6439 is approximately 50.

For a natural log, log_e(20) = ln(20) = 2.9957, because e^2.9957 is about 20. For a common log, log_10(1000) = 3, since 10^3 = 1000.

Key Logarithm Rules That Affect the Result

Knowing a few identities helps you sanity-check answers and simplify expressions before calculating:

• log_b(1) = 0 for any valid base, because b^0 = 1.
• log_b(b) = 1, since b^1 = b.
• Product rule: log_b(x·y) = log_b(x) + log_b(y).
• Quotient rule: log_b(x/y) = log_b(x) - log_b(y).
• Power rule: log_b(x^n) = n · log_b(x).

Common Mistakes and Practical Tips

The most frequent error is confusing log (base 10) with ln (base e). On many calculators "log" means base 10, so always confirm the base you actually want before reading the result.

Another mistake is entering a non-positive value. Because no real power of a positive base yields zero or a negative number, log_b(0) and logs of negatives have no real answer. Also remember the base cannot be 1, since 1 raised to any power is always 1.

If you only have base-10 or natural log buttons, use the change-of-base formula above — it works with either built-in log, and both give the same answer. Finally, watch significant figures: small rounding in ln(b) can shift the final digit, so keep extra decimals during intermediate steps.