What is a logarithm?

A logarithm is the inverse of an exponential function. If \[ b^y = x, \] with base \(b > 0\) and \(b \ne 1\), then the logarithm of \(x\) base \(b\) is \[ y = \log_b(x). \] In words: “\(y\) is the exponent you put on \(b\) to get \(x\)”.

Common choices of base are:

• Base 10: \(\log_{10}(x)\), often written simply as \(\log(x)\) in engineering.
• Base e: \(\log_e(x)\), written \(\ln(x)\), dominating calculus and analysis.
• Base 2: \(\log_2(x)\), fundamental in computer science and information theory.

Change-of-base formula

Calculators usually provide only \(\ln\) and \(\log_{10}\). To compute any base \(b\), you use the change-of-base formula:

As long as \(x > 0\), \(b > 0\) and \(b \ne 1\), the formula is valid and easy to evaluate on a scientific calculator or in software. Our interface uses this formula internally and shows you the intermediate values if you enable “show steps”.

Domain of the logarithm

For real-valued logarithms, there are strict domain rules:

• The argument must be positive: \(x > 0\).
• The base must be positive: \(b > 0\).
• The base cannot be 1: \(b \ne 1\).

If these conditions fail, the logarithm is not defined in the real numbers and the calculator will warn you instead of returning a misleading numeric result.

Examples

• \(\log_{10}(1000) = 3\) because \(10^3 = 1000\).
• \(\log_2(1024) = 10\) because \(2^{10} = 1024\).
• \(\log_5(125) = 3\) because \(5^3 = 125\).
• \(\ln(e^4) = 4\) because the natural log is the inverse of \(e^x\).

FAQ: using the logarithm calculator safely

Why does the calculator say my input is “out of domain”?

This happens when \(x \le 0\), \(b \le 0\) or \(b = 1\). In those cases, the logarithm is not defined in the real numbers. If you are working with complex analysis, you need a different tool that supports complex values.

Is there a difference between log and ln?

Yes. Strictly speaking, \(\ln(x)\) is log base \(e\), while \(\log(x)\) can mean log base 10 or base \(e\) depending on the context. This calculator always lets you pick the base explicitly, so you avoid ambiguity.

How many decimal places do I need?

For most homework and engineering applications, 4–6 decimal places are more than enough. You can increase the precision if you are propagating error through several calculations, but be careful not to over-interpret the extra digits.

Can I rely on this calculator for scientific or financial decisions?

This tool uses standard double-precision floating-point arithmetic, similar to a scientific calculator. It is suitable for educational and many practical uses, but you are ultimately responsible for checking whether the precision and model are appropriate for critical work (for example, safety calculations, regulatory reporting, or financial contracts).