Mathématiques et Algèbre

Logarithm & Exponential Solver — Calculate logᵦ(x), ln(x) & log₁₀(x)

Logarithms are foundational mathematical tools that invert exponential functions, answering the core question: 'To what exponent must a base be raised to produce a target number?' Our free Logarithm Calculator computes logarithms with any base, evaluates natural logarithms (ln), computes common logarithms (log₁₀), calculates binary logarithms (log₂), and solves exponential equations step-by-step. Equipped with rules cheat sheets and change of base formulas, it simplifies precalculus and computer science math.

Logarithm & Exponential Calculator

Calculate logᵦ(x), natural log ln(x), log₁₀(x) & solve exponential equations bˣ = y

Quick Base Presets:

Base (b)

Value (x)

Calculated Output

log₍10₎(100) = 2.000000

Step-by-Step Method: Change of base formula: log₍10₎(100) = ln(100) / ln(10) = 4.605170 / 2.302585 = 2.000000

Essential Logarithm Rules Cheat Sheet

Product Rule: logᵦ(x × y) = logᵦ(x) + logᵦ(y)

Quotient Rule: logᵦ(x / y) = logᵦ(x) - logᵦ(y)

Power Rule: logᵦ(xᵖ) = p × logᵦ(x)

Change of Base: logᵦ(x) = ln(x) / ln(b)

Mathematical Formulas & Calculation Methods

Logarithm Definition

A logarithm log_b(x) = y means that base b raised to the power y equals x.

log_b(x) = y ⇔ b^y = x

Change of Base Formula

Allows calculating logarithms of any base using natural logarithms (ln) or common logarithms (log₁₀).

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

Product & Quotient Rules

Expresses logs of products and quotients as sums and differences.

log_b(m · n) = log_b(m) + log_b(n) | log_b(m / n) = log_b(m) - log_b(n)

Power Rule of Logarithms

Brings exponents in the logarithm argument out front as multipliers.

log_b(x^k) = k · log_b(x)

Worked Step-by-Step Examples

Example 1: Evaluating log₂ (32)

"Find x for log₂(32) = x."

Steps:

Apply change of base formula: log₂(32) = ln(32) / ln(2).
Evaluate natural logs: ln(32) ≈ 3.465736, ln(2) ≈ 0.693147.
Divide values: 3.465736 / 0.693147 = 5.
Verify with exponentiation: 2⁵ = 32.

Result: log₂(32) = 5

Example 2: Solving Exponential Equation 5^x = 125

"Solve for x in the exponential equation 5^x = 125."

Steps:

Take natural log of both sides: ln(5^x) = ln(125).
Apply power rule: x · ln(5) = ln(125).
Isolate x: x = ln(125) / ln(5) = 4.8283 / 1.6094 = 3.
Verification: 5³ = 125.

Result: x = 3

What is Calculateur de Logarithm & Exponential?

A logarithm calculator is an online mathematical solver designed to evaluate logarithmic functions and solve exponential equations. The concept of logarithms was invented by John Napier in the 17th century to convert complex multiplication and division operations into simple addition and subtraction.

Logarithmic scales compress wide numerical ranges into manageable scales. Famous real-world applications include the pH scale for measuring acidity/alkalinity, the Richter scale for earthquake magnitude, decibels (dB) for sound intensity, and stellar magnitude in astronomy.

In mathematics, two logarithmic bases occur most frequently: the natural logarithm (ln) with base e (Euler's number ≈ 2.71828) used in calculus and exponential growth/decay models, and the common logarithm (log₁₀) used in general science and engineering.

How to Use Calculateur de Logarithm & Exponential

1Select your solver mode: Logarithm Solver (custom base b and value x) or Exponential Solver (b^x = y).
2Enter your base number b and target value x into the input boxes.
3Click calculate to view instant numerical results rounded to high precision.
4Examine the change of base breakdown and logarithmic rules applied during calculation.

Key Features

Custom base logarithm solver (supports non-standard positive bases).
One-click presets for Common Log (log₁₀), Natural Log (ln), and Binary Log (log₂).
Exponential equation solver (find x when b^x = y).
Step-by-step change of base formula expansion.
Interactive logarithmic identities and rules cheat sheet.

Benefits of Using Calculateur de Logarithm & Exponential

Eliminates the need for traditional printed logarithmic tables.

Solves tricky continuous compound interest and decay rate equations instantly.

Helps students master logarithmic rules for algebra and calculus exams.

Supports high-precision decimal outputs for scientific computations.

Practical Use Cases

Algebra & Precalculus Students

Solve exponential equations, logarithmic identities, and homework assignments.

Computer Science Students

Analyze algorithm time complexity (O(log n)) and binary search trees.

Chemistry & Physics Students

Calculate pH values, radioactive decay half-lives, and sound decibel levels.

Financial Analysts

Determine doubling times for investments using the Rule of 72 and continuous compounding.

Frequently Asked Questions

What is the difference between ln(x) and log(x)?

ln(x) represents the natural logarithm using base e (Euler's number ≈ 2.71828). log(x) typically represents the common logarithm using base 10 unless specified otherwise.

Why can't a logarithm base be equal to 1 or negative?

If base b = 1, 1 raised to any power is always 1, making log₁(x) undefined for any value other than 1. Negative bases create non-real complex outputs that alternate wildly across continuous inputs.

What is the change of base formula?

The change of base formula states that log_b(x) = ln(x) / ln(b) or log₁₀(x) / log₁₀(b). It enables calculating logarithms of any base on standard calculators.

What is the logarithm of 0 or a negative number?

In real-number mathematics, the logarithm of 0 and negative numbers is undefined because no positive real base b raised to any power can equal 0 or a negative value.

What is binary log (log₂)?

Binary log (log₂) is a logarithm with base 2. It is widely used in computer science to measure information entropy (bits), binary tree depth, and divide-and-conquer algorithms.

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