Log Calculator Calculate logarithms with any base, natural logs, and common logs — with step-by-step solutions and real-w...

Log Calculator

Calculate logarithms with any base, natural logs, and common logs — with step-by-step solutions and real-world applications.

Basic Logarithm

Change of Base

Log Properties

Antilogarithm

Number (x)

Base (b)

Log Type log_b(x) ln(x) log₁₀(x)

Number (x)

From Base

To Base

First Value

Second Value

Property Product (log(ab)) Quotient (log(a/b)) Power (log(a^n)) Root (log(ⁿ√a))

Exponent (n)

Log Value

Base (b)

Antilog Type b^x e^x 10^x

2

100

Number (x)

10

Base (b)

2

Logarithm

Common Log

Calculation Type

Logarithm Formulas

Basic Definition

log_b(x) = y ⇔ b^y = x

Change of Base

log_b(x) = log_k(x)/log_k(b)

Product Rule

log_b(xy) = log_b(x) + log_b(y)

Antilogarithm

antilog_b(y) = b^y

Interpretation

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Logarithmic Function Visualization

Logarithm Fundamentals

A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must we raise the base to get the number?"

Key Concepts**:

• Base (b)**: The number being raised to a power (must be positive and ≠ 1)
• Argument (x)**: The number we want to express as a power of the base (must be positive)
• Result (y)**: The exponent that satisfies b^y = x

Common Logarithms**:

• Common Log**: Base 10 (log₁₀ or just log)
• Natural Log**: Base e ≈ 2.71828 (ln)
• Binary Log**: Base 2 (log₂, used in computer science)

✅ Fundamental Relationship**: log_b(x) = y if and only if b^y = x.

Common Logarithm Mistakes

⚠️ Avoid these frequent errors:

• Log of non-positive numbers**: log(x) is undefined for x ≤ 0
• Invalid bases**: Base must be positive and not equal to 1
• Confusing ln and log**: ln is base e, log is usually base 10
• Product rule misuse**: log(a + b) ≠ log(a) + log(b)
• Negative results**: log(x) < 0 when 0 < x < 1 (for b > 1)

✅ Best Practices**:

• Always verify that your argument is positive before calculating
• Be explicit about your base when it's not obvious
• Use the change of base formula when your calculator doesn't support the desired base
• Remember that logarithms convert multiplication to addition

Real-World Applications

Logarithms are essential in:

• Science**: pH scale (acidity), Richter scale (earthquakes), decibels (sound)
• Finance**: Compound interest calculations, investment growth rates
• Computer Science**: Algorithm complexity (O(log n)), data structures
• Engineering**: Signal processing, control systems, information theory
• Mathematics**: Solving exponential equations, calculus, number theory

📊 Example Use Cases**:

• pH Calculation**: pH = -log₁₀[H⁺] → [H⁺] = 10^(-pH)
• Earthquake Magnitude**: Richter scale uses log₁₀ of wave amplitude
• Sound Intensity**: Decibels = 10 × log₁₀(I/I₀)
• Compound Interest**: Time to double = log(2)/log(1 + r)

How to Use This Calculator

➡️ Basic Logarithm

"log₁₀(100)" → 2 (because 10² = 100)

➡️ Change of Base

"Convert log₂(8) to base 10" → log₁₀(8)/log₁₀(2) = 3

➡️ Log Properties

"log(10 × 100)" → log(10) + log(100) = 1 + 2 = 3

➡️ Antilogarithm

"Antilog₁₀(2)" → 10² = 100

Note: All logarithmic calculations require positive arguments. Results are calculated with high precision using JavaScript's Math.log() function.