Log Calculator

What this log calculator does

A logarithm answers the question: to what power must the base be raised to produce a given value? This calculator evaluates logarithms for a positive value and a base you choose. Use e for the natural logarithm and 10 for the common logarithm.

For example, log₁₀(1000) = 3 because 10³ = 1000.

How to use this calculator

1. Enter a positive value.
2. Enter the base, or e for the natural logarithm.
3. Click Calculate.

The default example is ln(100), which is about 4.6052.

Logarithm formulas

y = log_b(x) means b^y = x
log_b(x) = ln(x) / ln(b)
ln(x) = log_e(x)
log10(x) = log_10(x)

The value must be positive. The base must be positive and not equal to 1.

What the results mean

The result is the exponent y. For ln(100), the result is about 4.60517, meaning e^4.60517 ≈ 100.

A logarithm of a number between 0 and 1 is negative. A logarithm of 1 is zero for any valid base.

Assumptions and limitations

The calculator accepts real positive values only. It does not return complex logarithms for negative values or non-positive bases. It uses standard floating-point precision.

It does not simplify symbolic logarithms or apply logarithm identities beyond the numeric evaluation.

Frequently asked questions

What is the difference between ln and log?

ln is the natural logarithm with base e. log often means base 10 in school settings, but in higher mathematics it may mean the natural logarithm. This calculator makes the base explicit.

Why can’t I take the log of a negative number?

In real numbers, a positive base raised to any real power is positive, so a negative result is impossible. Complex logarithms require a different treatment.

Why can’t the base be 1?

1 raised to any power is 1, so it cannot produce a range of values. The base must be positive and not equal to 1.

What is log(1)?

For any valid base, log_b(1) = 0 because b^0 = 1.

Source

Reference: Wolfram MathWorld — Logarithm.

Disclaimer

This calculator is for educational purposes. Verify high-precision results with an appropriate math tool.