Interactive mathematics

Logarithm Calculator

Find a logarithm, its argument or its base. Explore the inverse power equation and see the answer on a graph that follows your values.

Three solving modes · responsive graph · worked formulas

What would you like to find?

Base bPositive and not 1

Argument xPositive number

Enter decimals, scientific notation, e, or a simple fraction such as 1/2.

Calculated answer

log₂(8) = 3

Check with the inverse power2 ^ 3 = 8

Change-of-base calculationln(8) / ln(2) ≈ 3

The point marks your calculation. The curve rises when b > 1 and falls when 0 < b < 1.

Your point(1, 0)(b, 1)

Vertical asymptote: x = 0 · Real-valued graph: x > 0

A logarithm asks which power of b gives x.

The statements log_b(x) = y and b^y = x describe the same relationship. Use the inverse form to check a result.

log_b(x) = y ⇔ b^y = x

Use ln(x) / ln(b) to calculate a logarithm with any positive base other than 1.

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

For real values, x must be positive, b must be positive and b cannot equal 1. Negative logarithm results are valid.

x > 0 · b > 0 · b ≠ 1

For a base between 0 and 1, increasing x makes the logarithm decrease. The graph must show this change of direction.

Continue with powers, roots and scientific notation.

What does log_b(x) = y mean?

It means b raised to the power y equals x.

Can the logarithm be negative?

Yes. For example, log_2(1/2) = -1 because 2^-1 = 1/2.

What bases are allowed?

Any positive base except 1. A base between 0 and 1 produces a decreasing logarithm graph.

Can I enter a fraction?

Yes. Enter a simple fraction such as 1/2, or use a decimal or scientific notation.

Why is the result sometimes approximate?

Most logarithms are not terminating decimals. The calculator displays a numerical approximation and the inverse check.

Can I solve for the base or argument?

Yes. Choose the unknown quantity first, then enter the other two values.