Logarithm Calculator (Free Log & Antilog Solver with Steps)

Q: What is the formula for calculating a logarithm?

The logarithm log_b(x) answers the fundamental question: 'What exponent do we need to raise base b to in order to get x?'. Mathematically, if y = log_b(x), then b^y = x. To compute arbitrary bases, use the Change of Base formula: log_b(x) = ln(x) / ln(b).

Q: How do I calculate a logarithm with a custom base?

To calculate a custom base logarithm like log_2(8) without a specialized scientific calculator button, apply the Change of Base formula: log_b(x) = ln(x) / ln(b) or log10(x) / log10(b). In our calculator, simply enter your custom base b and argument x, and our solver automatically applies this formula and displays the step-by-step division.

Q: What is the difference between log and ln?

'log' typically refers to the Common Logarithm, which uses base 10 (log10). 'ln' refers to the Natural Logarithm, which uses Euler's number (e ≈ 2.71828) as its base. In science and engineering, ln is widely used for continuous growth and decay models.

The most accurate online log calculator and logarithms solver. Compute common log (base 10), natural log (ln), custom bases, and inverse antilogarithms with instant step-by-step mathematical derivations. Zero servers, 100% private.

log ( ) = 2

Exponential Verification Proof

10² = 100

Step-by-Step Mathematical Derivation

How to Calculate Logarithms Online (Step-by-Step)

Whether you are evaluating a basic common logarithm or solving a complex equation with a custom base, this calculator with logs automates the entire derivation in three simple steps:

Select Logarithm Mode or Quick Base

Choose between Logarithm mode (log_b(x)) or Antilog (Inverse) mode (b^y). Use the quick toggles to instantly set Base 10 (common log), Base e (natural log ln), or Base 2 (binary log).

Enter the Base and Argument

Input your desired base number b (must be greater than 0 and not equal to 1) and your argument number x (must be strictly positive). For antilog equations, input the base b and exponent y.

Review the Step-by-Step Derivation and Proof

The tool instantly renders the exact calculation steps using the Change of Base formula alongside the exponential verification proof (b^y = x). Click "Copy Steps" to copy the formatted derivation into your homework or notes.

Why You Need The Change of Base Formula

The Problem (Premise): Most standard scientific calculators, Excel formulas, and programming language math libraries only provide built-in functions for the Common Logarithm (log₁₀) and Natural Logarithm (ln or base e). They lack a dedicated button to directly compute custom bases such as log₂(8) or log₅(125).

The Mathematical Solution (Evidence): To calculate a logarithm with an arbitrary custom base, you must apply the universal Change of Base Formula, which converts any base into a ratio of natural or common logarithms:

log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)

Automated Execution (Conclusion): Our logarithm calculator automates this conversion behind the scenes. When you enter any custom base, our algorithmic engine performs the precision ratio calculation and presents the step-by-step division so you can verify the math instantly without manual fraction division.

Quick Reference Table of Common Logarithms

For rapid reference in mathematics, physics, and computer science, here is a structured comparison chart of standard logarithmic values across Base 10 (Common Log), Base e (Natural Log ln), and Base 2 (Binary Log):

Argument (x)	Common Log (log₁₀ x)	Natural Log (ln x)	Binary Log (log₂ x)	Exponential Meaning
1	0	0	0	Any base raised to 0 equals 1 (b⁰ = 1)
2	0.30103	0.69315	1	2¹ = 2
e (≈ 2.71828)	0.43429	1	1.44270	e¹ = e
10	1	2.30259	3.32193	10¹ = 10
100	2	4.60517	6.64386	10² = 100
1,000	3	6.90776	9.96578	10³ = 1,000
1,024	3.01030	6.93147	10	2¹⁰ = 1,024 (Kilobyte boundary in computing)

Developer Snippet: Programmatic Logarithms in JS & Python

If you are building software or scripts that require custom-base logarithmic calculations, you must implement the Change of Base formula programmatically. Below are production-ready, copy-pasteable snippets in JavaScript and Python:

JavaScript / TypeScript Math.log()

/**
 * Calculates custom base logarithm: log_b(x)
 * @param {number} x - Argument (must be > 0)
 * @param {number} b - Base (must be > 0 and !== 1)
 */
function getCustomLog(x, b) {
  if (x <= 0 || b <= 0 || b === 1) {
    throw new Error("Invalid logarithm inputs.");
  }
  // Change of Base: ln(x) / ln(b)
  return Math.log(x) / Math.log(b);
}

// Example: log_2(1024) = 10
console.log(getCustomLog(1024, 2)); // 10

Python 3.x math.log(x, base)

import math

def get_custom_log(x: float, base: float) -> float:
    """Calculates logarithm of x to the given base."""
    if x <= 0 or base <= 0 or base == 1:
        raise ValueError("Invalid logarithm inputs.")
    # Python math.log supports base natively
    return math.log(x, base)

# Example: log_2(1024) = 10.0
print(get_custom_log(1024, 2)) # 10.0

Logarithm Rules & Laws Reference

When algebraic equations involve logarithmic terms, the following universal mathematical identities (laws of logarithms) allow you to simplify, expand, or condense expressions:

Product Rule

log_b(x × y) = log_b(x) + log_b(y)

The logarithm of a multiplied product of two numbers is equal to the sum of their individual logarithms.

Quotient Rule

log_b(x / y) = log_b(x) - log_b(y)

The logarithm of a divided quotient is equal to the difference between the logarithm of the numerator and denominator.

Power Rule

log_b(x^p) = p × log_b(x)

An exponent on the argument can be moved directly to the front of the logarithm as a scalar multiplier.

Change of Base Rule

log_b(x) = log_c(x) / log_c(b)

Allows evaluating any custom base b by dividing logarithms of a convenient known base c (such as base 10 or base e).

Frequently Asked Questions

What is the formula for calculating a logarithm?

The logarithm log_b(x) answers the fundamental question: "What exponent do we need to raise base b to in order to get the number x?". Mathematically, if y = log_b(x), then b^y = x. To calculate arbitrary bases on standard calculators or computers, use the Change of Base formula: log_b(x) = ln(x) / ln(b).

How do I calculate a logarithm with a custom base?

To calculate a custom base logarithm like log₂(8) without a specialized scientific calculator button, apply the Change of Base formula: log_b(x) = ln(x) / ln(b) or log₁₀(x) / log₁₀(b). In our logarithms calculator, simply enter your custom base b and argument x, and our solver automatically applies this formula and displays the step-by-step division.

Can I calculate the log of a negative number or zero?

No, within the realm of Real Numbers, you cannot take the logarithm of zero or a negative number. The argument x must be strictly greater than 0. Raising a positive base (like 10 or e) to any real exponent always produces a strictly positive result. Attempting log(-5) or log(0) in our calculator will gracefully trigger a validation error rather than returning `NaN`.

What is the difference between log and ln?

log typically refers to the Common Logarithm, which uses base 10 (log₁₀). It is standard in chemistry (pH scales), acoustics (decibels), and earthquake measurement (Richter scale). ln refers to the Natural Logarithm, which uses Euler's number (e ≈ 2.71828) as its base. In calculus, physics, and continuous finance, ln is the natural choice for continuous growth and decay models.

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