Log Calculator

Calculate logarithms, natural logarithms, common logarithms, antilogarithms, and change of base with our free log calculator

Available Calculators

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. If bˣ = y, then log_b(y) = x. Logarithms are fundamental in mathematics, science, and engineering, used for solving exponential equations, calculating growth rates, and working with very large or very small numbers.

Logarithm Properties

Key properties of logarithms include:

_b(xy) = _b(x) + _b(y) \ _b(x/y) = _b(x) - _b(y) \ _b(x^n) = n _b(x)

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another: log_b(x) = log_c(x) / log_c(b). This is particularly useful when working with calculators that only support natural log (ln) or common log (log₁₀).

Examples of Logarithm Calculations

Here are several everyday situations that show how easy it is to use our log calculator.

Example 1: Calculate log₁₀(100)

Find the common logarithm of 100.

You can use our calculator by entering:

What is log₁₀(100)?

The calculator returns: 2

log₁₀(100) = ?
10² = 100
Therefore, log₁₀(100) = 2

The formula used is:

_10(100) = 2

Example 2: Natural Log of e

Calculate the natural logarithm of e.

You can use our calculator by entering:

What is ln(e)?

The calculator returns: 1

ln(e) = ?
e¹ = e
Therefore, ln(e) = 1

The formula used is:

(e) = 1

Example 3: Logarithm with Base 2

Calculate log₂(8).

You can use our calculator by entering:

What is log₂(8)?

The calculator returns: 3

log₂(8) = ?
2³ = 8
Therefore, log₂(8) = 3

The formula used is:

_2(8) = 3

Example 4: Antilog Calculation

Calculate 10³.

You can use our calculator by entering:

What is 10³?

The calculator returns: 1000

10³ = 10 × 10 × 10
10³ = 1000

The formula used is:

10^3 = 1000

Example 5: Change of Base

Convert log₁₀(100) to base 2.

You can use our calculator by entering:

What is log₂(100)?

The calculator returns: ≈ 6.644

log₂(100) = log₁₀(100) / log₁₀(2)
log₂(100) = 2 / 0.301
log₂(100) ≈ 6.644

The formula used is:

_2(100) = (_10(100) ÷ _10(2)) = (2 ÷ 0.301) ≈ 6.644

Example 6: Natural Log of 2.718

Calculate ln(2.718).

You can use our calculator by entering:

What is ln(2.718)?

The calculator returns: ≈ 1.000

ln(2.718) ≈ 1.000
Since e ≈ 2.718, ln(2.718) ≈ 1

The formula used is:

(2.718) ≈ 1.000

Example 7: Common Log of 1000

Calculate log₁₀(1000).

You can use our calculator by entering:

What is log₁₀(1000)?

The calculator returns: 3

log₁₀(1000) = ?
10³ = 1000
Therefore, log₁₀(1000) = 3

The formula used is:

_10(1000) = 3

Example 8: Logarithm of Product

Calculate log₁₀(100 × 10).

You can use our calculator by entering:

What is log₁₀(1000)?

The calculator returns: 3

log₁₀(100 × 10) = log₁₀(100) + log₁₀(10)
log₁₀(100 × 10) = 2 + 1 = 3

The formula used is:

_10(100 × 10) = _10(100) + _10(10) = 2 + 1 = 3

Example 9: Antilog of 2

Calculate 10².

You can use our calculator by entering:

What is 10²?

The calculator returns: 100

10² = 10 × 10
10² = 100

The formula used is:

10^2 = 100

Example 10: Natural Log of 1

Calculate ln(1).

You can use our calculator by entering:

What is ln(1)?

The calculator returns: 0

ln(1) = ?
e⁰ = 1
Therefore, ln(1) = 0

The formula used is:

(1) = 0

Summary

These examples show how our log calculator can handle:

  • •Calculating logarithms with any base (log₁₀(100) = 2)
  • •Natural logarithms (ln(e) = 1)
  • •Common logarithms (log₁₀(1000) = 3)
  • •Antilogarithms and change of base

You can adapt any real-life situation involving logarithms to one of these patterns and let the calculator do the work for you.

How to Use

  1. Enter the value and base (if applicable) for the logarithm calculation.
  2. Select the type of logarithm calculation you need (log, ln, log₁₀, antilog, or change of base).
  3. View the result and step-by-step calculation breakdown.

Frequently asked questions

What is a logarithm?

A logarithm is the inverse operation of exponentiation. If bˣ = y, then log_b(y) = x.

What is the difference between natural log and common log?

Natural log (ln) uses base e (≈2.718), while common log (log₁₀) uses base 10. Natural log is common in calculus and science, while common log is used in engineering and everyday calculations.

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About this tool

Published by Online Tools Center

Last updated:

Check the units, assumptions and rounding before using a result. Health and financial estimates are educational and do not replace individual professional advice.

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