Log Calculator - Calculate Logarithm, Natural Log, Antilog Online

Our free online log calculator helps you calculate logarithms, natural logarithms, and antilogarithms with different bases. Whether you need to find the logarithm of a number, calculate natural log (ln), common log (log₁₀), or antilogarithm, our tool provides accurate results with step-by-step calculations.

What is a Logarithm?

A logarithm is the inverse operation of exponentiation. If \(a^b = c\), then \(\log_a(c) = b\). In other words, the logarithm of a number is the exponent to which another fixed value (the base) must be raised to produce that number.

Types of Logarithms

Common Logarithm (log₁₀): Base 10 logarithm, widely used in science and engineering
Natural Logarithm (ln): Base e logarithm, where e ≈ 2.71828, commonly used in mathematics and physics
Binary Logarithm (log₂): Base 2 logarithm, used in computer science and information theory
Custom Base Logarithm: Logarithm with any positive base (except 1)

Logarithm Formulas

Basic Logarithm Formula

For any positive numbers \(a\), \(b\), and \(c\) where \(a \neq 1\):

\[\log_a(b) = c \text{ if and only if } a^c = b\]

Change of Base Formula

To convert between different logarithm bases:

\[\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\]

Where \(c\) is any positive number (commonly 10 or e).

Natural Logarithm Properties

\(\ln(1) = 0\)
\(\ln(e) = 1\)
\(\ln(ab) = \ln(a) + \ln(b)\)
\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
\(\ln(a^n) = n\ln(a)\)

Common Logarithm Values

Number	log₁₀	ln	log₂
1	0	0	0
2	0.3010	0.6931	1
e	0.4343	1	1.4427
10	1	2.3026	3.3219
100	2	4.6052	6.6439

How to Use the Log Calculator

Select Operation Type: Choose between logarithm, natural logarithm, common logarithm, or antilogarithm
Choose Base: Select the base for your calculation (2, 10, e, or custom)
Enter Value: Input the number you want to calculate the logarithm for
Get Results: View the calculated result and step-by-step solution

Applications of Logarithms

Mathematics

Solving exponential equations
Calculus and differential equations
Complex number analysis
Probability and statistics

Science and Engineering

pH calculations in chemistry
Decibel measurements in acoustics
Radioactive decay calculations
Signal processing and filtering

Computer Science

Algorithm complexity analysis
Information theory
Data compression
Cryptography

Finance and Economics

Compound interest calculations
Population growth models
Economic forecasting
Risk assessment

Frequently Asked Questions

What is the difference between log and ln?

log typically refers to base 10 logarithm (log₁₀), while ln refers to natural logarithm (base e). The natural logarithm is more commonly used in advanced mathematics and physics.

Can I calculate logarithm of negative numbers?

No, logarithms of negative numbers are not defined in real numbers. The logarithm function is only defined for positive real numbers.

What is the logarithm of zero?

The logarithm of zero is undefined. As you approach zero from the positive side, the logarithm approaches negative infinity.

How do I convert between different logarithm bases?

Use the change of base formula: \(\log_a(b) = \frac{\log_c(b)}{\log_c(a)}\), where \(c\) is any positive base (commonly 10 or e).

What is antilogarithm?

Antilogarithm is the inverse operation of logarithm. If \(\log_a(b) = c\), then the antilogarithm of \(c\) with base \(a\) is \(b = a^c\).

Why is e used as the base for natural logarithm?

The number e (≈ 2.71828) is a special mathematical constant that arises naturally in many areas of mathematics, particularly in calculus. The natural logarithm has elegant properties that make it ideal for mathematical analysis.

How accurate are the calculations?

Our calculator provides results with high precision (up to 10 decimal places) using JavaScript's built-in Math functions, which are accurate for most practical purposes.

Can I use this calculator for complex numbers?

This calculator is designed for real numbers only. For complex number logarithms, you would need specialized mathematical software.

Tips for Using Logarithms

Always ensure your input values are positive
Remember that \(\log(1) = 0\) for any base
Use the change of base formula when working with different bases
For very large or very small numbers, consider using scientific notation
Double-check your calculations, especially when working with logarithms in equations

Mathematical Properties

Logarithm Rules

\(\log_a(1) = 0\) for any base \(a\)
\(\log_a(a) = 1\) for any base \(a\)
\(\log_a(ab) = \log_a(a) + \log_a(b)\)
\(\log_a\left(\frac{a}{b}\right) = \log_a(a) - \log_a(b)\)
\(\log_a(a^n) = n\)
\(\log_a(b^n) = n\log_a(b)\)

Special Values

\(\log_{10}(10) = 1\)
\(\log_{10}(100) = 2\)
\(\log_{10}(1000) = 3\)
\(\ln(e) = 1\)
\(\ln(1) = 0\)
\(\log_2(2) = 1\)
\(\log_2(4) = 2\)
\(\log_2(8) = 3\)

Our log calculator is a powerful tool for students, professionals, and anyone working with logarithmic functions. It provides accurate calculations with detailed step-by-step solutions, making it perfect for learning and verification purposes.