online log calculator

What this online log calculator does

This tool calculates three values in one click: the logarithm of x in your chosen base b, the natural logarithm ln(x), and the common logarithm log₁₀(x). It is useful for algebra, statistics, data science, finance, and exam prep.

Instead of manually using change-of-base every time, you can enter your number once and get clean, formatted results. The calculator also validates inputs so you avoid undefined results.

How to use it

1. Enter the number you want to evaluate in the Number (x) field.
2. Enter the logarithm base in the Base (b) field.
3. Choose how many decimal places you want.
4. Click Calculate.

If you enter invalid values (such as a negative number or base 1), the tool will explain exactly what to fix.

Logarithm basics (quick refresher)

A logarithm answers this question: “To what exponent must I raise base b to get x?”

In equation form:

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

Common types of logs

• Natural log: ln(x) = log_e(x), where e ≈ 2.71828
• Common log: log(x) usually means log₁₀(x)
• Binary log: log₂(x), common in computer science

Examples

Example 1: log₂(64)

Because 2⁶ = 64, the answer is 6.

Example 2: log₁₀(1000)

Because 10³ = 1000, the answer is 3.

Example 3: ln(20)

The value is approximately 2.995732. This appears often in growth and decay models.

Where logarithms are used in real life

• Finance: compound growth and continuously compounded returns
• Science: pH scale, sound intensity (decibels), earthquake magnitude
• Machine learning: log loss, entropy, and likelihood functions
• Engineering: Bode plots and signal analysis
• Statistics: normalizing skewed data with log transforms

Important input rules

• The number x must be greater than zero.
• The base b must be greater than zero.
• The base b cannot equal 1, because that makes the logarithm undefined.

FAQ

Why can’t I take the log of 0 or a negative number?

For real-number logarithms, the domain is strictly positive numbers. So x must be > 0.

What if I need more precision?

Increase the decimal places up to 15. For very large or tiny outputs, scientific notation may be shown.

How is log_b(x) calculated?

This calculator uses the change-of-base identity: log_b(x) = ln(x) / ln(b).