Log Calculator (Logarithm Calculator Online)

What is a Logarithm? (Logarithm Basics Explained)

A logarithm is the inverse of exponentiation. In simple terms, a logarithm answers the question: “To what power must a base be raised to produce a given number?” For example, log₁₀(100) = 2 because 10² = 100.

There are three common types of logarithms:

• Common Logarithm (base 10): log₁₀

• Natural Logarithm (base e): ln

• Binary Logarithm (base 2): log₂

Logarithms are used in solving logarithmic equations, exponential functions, and real-world problems such as measuring earthquake magnitude or sound intensity.

Understanding logarithms also requires knowing their rules and properties, such as the product rule, quotient rule, and power rule. These rules allow complex expressions to be simplified.

In math, logs are crucial because they transform multiplication into addition and division into subtraction, making large calculations manageable. Whether you’re a student asking “What is log in math?” or a professional calculating exponential growth, knowing how logs work is essential.

How to Use the Log Calculator Online

Using our Logarithm Calculator Online is quick and simple. Follow these steps:

1. Enter the number you want to calculate.

2. Choose the base (10, e, 2, or custom).

3. Click Calculate, and the tool instantly shows the result.

For example:

• Input: Number = 1000, Base = 10 → Result: 3 (since 10³ = 1000).

• Input: Number = 8, Base = 2 → Result: 3 (since 2³ = 8).

This log calculator with steps also explains the process, making it useful for students learning logarithm formulas. It works as a natural logarithm calculator online (ln calculator), a log base 10 calculator, or even a logarithm solver for custom bases.

Whether you’re asking “How to calculate logarithm?” or “What’s the difference between log and ln?”, this tool provides the solution.

👉 Want to explore more? Check out our Roots Calculator.

Logarithm Rules and Properties

To effectively use a logarithm calculator, you need to understand basic logarithm rules. These rules allow you to simplify complex logarithmic expressions into manageable forms.

1. Product Rule: logₐ(xy) = logₐ(x) + logₐ(y)

2. Quotient Rule: logₐ(x/y) = logₐ(x) – logₐ(y)

3. Power Rule: logₐ(xⁿ) = n × logₐ(x)

4. Change of Base Formula: logₐ(x) = log_b(x) / log_b(a)

For example:

• log₁₀(1000) = log₁₀(100 × 10) = log₁₀(100) + log₁₀(10) = 2 + 1 = 3

• log₂(64) = log₂(2⁶) = 6 × log₂(2) = 6

These rules are the foundation of logarithmic problem solving in algebra, calculus, and computer science.

👉 For a complete guide, visit Khan Academy’s Logarithm Rules.

By applying these properties, our logarithm calculator with solution makes learning and problem-solving easier.

Types of Log Calculators (Base 10, Base e, Base 2, Antilog)

Our tool functions as multiple calculators in one:

• Log Base 10 Calculator (Common Log): Widely used in science and engineering. Example: log₁₀(1000) = 3.

• Natural Log Calculator (ln, Base e): Important in calculus, growth, and decay models. Example: ln(e²) = 2.

• Log Base 2 Calculator (Binary Log): Essential in computing, data compression, and algorithms. Example: log₂(32) = 5.

• Antilog Calculator (Inverse Log): Used to reverse logarithmic functions. Example: If log₁₀(x) = 3, then antilog₁₀(3) = 1000.

Each type has its own applications, but they share the same principles. This is why our logarithm calculator online is versatile—allowing you to switch between bases without needing multiple tools.

Whether you’re solving a binary logarithm, working on exponential equations, or simply exploring “what is natural log,” this calculator saves time and ensures accuracy.

Examples of Logarithm Calculations with Steps

Let’s go through some examples to see how the log calculator with solution works:

1. Example 1: log₁₀(1000) = ?
   → Since 10³ = 1000, the answer is 3.

2. Example 2: ln(e²) = ?
   → Since e² = e × e, the result is 2.

3. Example 3: log₂(16) = ?
   → Since 2⁴ = 16, the answer is 4.

4. Example 4: log₅(25) = ?
   → Since 5² = 25, the result is 2.

These worked examples demonstrate how logarithms reduce exponentiation into simpler arithmetic. The logarithm calculator with steps provides similar detailed solutions, helping students understand the process instead of just the final answer.

👉 For deeper reference, check Wolfram MathWorld: Logarithm.