Log Calculator – Log Base 10, Natural Log, and Any Base Calculations

  • Adam
  • November 4, 2025

Free online log calculator for computing common logarithms, natural logarithms, logarithms with any base, antilog, and change of base conversions. Step-by-step solutions for algebra, calculus, and scientific applications.

Log Calculator

Advanced Tool for Computing Logarithms and Exponential Relationships

Quick Navigation

Common Logarithm (Base 10)

Calculate log₁₀(x)

Natural Logarithm (Base e)

Calculate ln(x) = log_e(x)

Logarithm with Any Base

Calculate log_b(x) for any base b

Antilog (Inverse Logarithm)

Calculate antilog_b(y) = b^y

Change of Base Formula

Convert log_b(x) to different base

What are Logarithms?

Logarithms are mathematical operations that are the inverse of exponentiation. If we say "2 raised to the power 3 equals 8" (2³ = 8), the logarithmic equivalent is "the logarithm base 2 of 8 is 3" (log₂(8) = 3). Logarithms answer the question: "To what power must we raise this base to get that number?"

The three main types of logarithms are: Common logarithm (base 10, written as log(x)), Natural logarithm (base e, written as ln(x)), and logarithms with any arbitrary base. Logarithms are fundamental in mathematics, appearing in scientific calculations, computer science, and countless real-world applications.

Logarithms transform exponential relationships into linear ones, which simplifies calculations and makes patterns easier to understand. They're particularly useful for dealing with very large or very small numbers, exponential growth and decay, and solving exponential equations.

Key Concept: Logarithms and exponents are inverse operations. If log_b(x) = y, then b^y = x. They undo each other.

Key Features & Capabilities

This comprehensive logarithm calculator provides multiple calculation modes and detailed analysis:

📊 Common Logarithm Calculate log₁₀(x) for base 10 logarithms used in science and engineering
🔢 Natural Logarithm Calculate ln(x) with base e (Euler's number) for calculus and advanced mathematics
📐 Any Base Calculate logarithm for any positive base value using the universal logarithm formula
⬆️ Antilog Calculation Find the inverse: calculate b^y when you know the logarithm value
🔄 Change of Base Convert between different logarithmic bases using the change of base formula
📋 Step-by-Step Solutions See detailed breakdown showing the calculation method and formula application
🧮 Related Values Display complementary values and verify relationships between logarithms
✓ Multiple Formats Results displayed in decimal, approximation, and exact forms
🎓 Educational Content Comprehensive guides and examples explaining logarithmic concepts
📋 Copy to Clipboard One-click copy functionality to transfer results to other applications
⚡ Real-Time Calculation Instant results with no delays or external dependencies
📱 Fully Responsive Works seamlessly on desktop, tablet, and mobile devices

How to Use This Calculator

Step-by-Step Guide

  1. Choose Your Logarithm Type: Select the appropriate tab: Log Base 10 for common logarithms, Natural Log for ln(x), Any Base for custom bases, Antilog for inverse operations, or Change Base for conversions.
  2. Enter Your Values: Input the required numbers. For most logarithms, enter the number you want the logarithm of. For antilog, enter the exponent value. The number must be positive.
  3. Specify Base if Needed: For "Any Base" mode, enter the base value. For "Antilog," select the base or enter a custom base.
  4. Click Calculate: Press the Calculate button to perform the computation using the appropriate logarithm formula.
  5. Review Results: The main result displays prominently showing the calculated logarithm value in decimal form.
  6. Study the Steps: Below the result, see detailed breakdown showing the formula used, calculation method, and how the answer was obtained.
  7. Analyze Statistics: See related values and verifications that help understand the logarithmic relationship.
  8. Copy or Clear: Use Copy to transfer results. Use Clear to reset for a new calculation.

Tips for Accurate Use

  • Positive Numbers Only: Logarithms are only defined for positive numbers. The input must always be greater than zero.
  • Valid Bases: The base must be a positive number not equal to 1. Common bases are 10, e, and 2.
  • Decimal Support: All calculators support decimal inputs for precise calculations.
  • Verification: Use antilog to verify: if log_b(x) = y, then antilog_b(y) should equal x.
  • Base Relationships: Remember that changing the base uses the change of base formula: log_a(x) = log_b(x) / log_b(a).

Complete Formulas Guide

Logarithm Definition

Basic Logarithm Relationship
If b^y = x, then log_b(x) = y

Where:
b = base (b > 0, b ≠ 1)
x = argument (x > 0)
y = logarithm result

Example: 2^3 = 8, so log₂(8) = 3

Common Logarithm

Logarithm Base 10
log(x) = log₁₀(x)

Used in: pH calculations, decibel measurements, Richter scale

Examples:
log₁₀(1) = 0 (because 10^0 = 1)
log₁₀(10) = 1 (because 10^1 = 10)
log₁₀(100) = 2 (because 10^2 = 100)

Natural Logarithm

Logarithm Base e
ln(x) = log_e(x)

Where e ≈ 2.71828 (Euler's number)
Used in: calculus, compound interest, exponential growth/decay

Examples:
ln(1) = 0 (because e^0 = 1)
ln(e) = 1 (because e^1 = e)
ln(e^2) = 2 (because e^2 = e²)

Change of Base Formula

Converting Between Bases
log_a(x) = log_b(x) / log_b(a)

This allows converting any logarithm to base 10 or base e:
log_a(x) = ln(x) / ln(a)
log_a(x) = log(x) / log(a)

Example: Convert log₂(8) to base 10
log₂(8) = log(8) / log(2) = 0.903 / 0.301 = 3

Antilog (Inverse Logarithm)

Finding the Argument
If log_b(x) = y, then x = antilog_b(y) = b^y

For base 10: antilog(y) = 10^y
For base e: antilog(y) = e^y = exp(y)

Example: If log₁₀(x) = 2, then x = 10^2 = 100

Logarithm Properties

Product Property

log_b(xy) = log_b(x) + log_b(y) - The logarithm of a product equals the sum of logarithms. This converts multiplication into addition.

Quotient Property

log_b(x/y) = log_b(x) - log_b(y) - The logarithm of a quotient equals the difference of logarithms. This converts division into subtraction.

Power Property

log_b(x^n) = n · log_b(x) - The logarithm of a power equals the exponent times the logarithm of the base. This moves exponents out as multipliers.

Base Equivalence

log_b(b) = 1 - The logarithm of the base itself always equals 1, and log_b(1) = 0 - the logarithm of 1 is always 0.

Inverse Properties

b^(log_b(x)) = x and log_b(b^x) = x - These show the inverse relationship between logarithms and exponents.

Worked Examples

Example 1: Common Logarithm

Problem: Calculate log₁₀(1000)

Solution:
We need to find: "10 to what power equals 1000?"
10^? = 1000
10^3 = 1000

Therefore: log₁₀(1000) = 3

Verification: 10^3 = 10 × 10 × 10 = 1000 ✓

Example 2: Natural Logarithm

Problem: Calculate ln(e²)

Solution:
We need to find: "e to what power equals e²?"
e^? = e²
e^2 = e²

Therefore: ln(e²) = 2

Verification: e^2 = e² ✓

Example 3: Logarithm with Any Base

Problem: Calculate log₂(32)

Solution:
We need to find: "2 to what power equals 32?"
2^? = 32
2^5 = 32

Therefore: log₂(32) = 5

Verification: 2^5 = 2×2×2×2×2 = 32 ✓

Example 4: Change of Base

Problem: Convert log₂(8) using change of base to base 10

Solution:
Using change of base: log_a(x) = log_b(x) / log_b(a)
log₂(8) = log₁₀(8) / log₁₀(2)
= 0.903 / 0.301
= 3

Direct verification: log₂(8) = 3 because 2^3 = 8 ✓

Example 5: Antilog

Problem: Find x if log₁₀(x) = 2

Solution:
If log₁₀(x) = 2, then x = antilog₁₀(2) = 10^2
x = 100

Verification: log₁₀(100) = log₁₀(10²) = 2 ✓

Frequently Asked Questions

❓ Why are logarithms useful?
Logarithms convert exponential relationships into linear ones, making calculations easier. They handle very large/small numbers compactly, simplify multiplication (adding logs), and are essential for solving exponential equations. They appear in pH, decibels, earthquake scales, and financial calculations.
❓ What's the difference between log and ln?
log (or log₁₀) is base 10, used in science and engineering. ln is natural logarithm (base e), used in calculus and advanced mathematics. Both follow identical properties; only the base changes. Most scientific calculators have both.
❓ Can I take log of negative numbers?
No. Logarithms are undefined for negative numbers and zero in real numbers. You can only take logarithms of positive numbers. For negative inputs, you'd need complex numbers and get complex results.
❓ What is e and why is it important?
e (approximately 2.71828) is Euler's number, a fundamental constant in mathematics. It appears naturally in compound interest, exponential growth, and calculus. The natural logarithm (base e) is particularly useful in higher mathematics.
❓ Why is log₁₀(10) = 1?
Because 10^1 = 10. By definition, if log_b(x) = y, then b^y = x. So log₁₀(10) = 1 because 10 to the power 1 equals 10. Similarly, log_b(b) always equals 1.
❓ What does log₁₀(1) equal?
log₁₀(1) = 0 because 10^0 = 1. In fact, for any valid base: log_b(1) = 0. This is because any number to the zero power equals 1.
❓ How do logarithms and exponents relate?
They're inverse operations. If log_b(x) = y, then b^y = x. They undo each other: b^(log_b(x)) = x and log_b(b^x) = x. Understanding this relationship is key to logarithms.
❓ Can the base of a logarithm be 1?
No. If base = 1, then 1 raised to any power equals 1, making the logarithm ambiguous and undefined. The base must be a positive number not equal to 1.
❓ What are practical uses of logarithms?
Logarithms appear in: pH calculations in chemistry, decibel measurements in acoustics, Richter scale for earthquakes, time complexity in algorithms, compound interest in finance, exponential decay in physics, and radiocarbon dating in archaeology.
❓ When would I use this calculator?
Use for: solving math problems, verifying calculations, understanding logarithmic concepts, chemistry (pH calculations), physics (exponential decay), finance (compound interest), computer science (algorithmic analysis), or any application involving logarithms or exponential relationships.

Start Calculating Logarithms

Whether you're solving algebra problems, analyzing exponential relationships, working with scientific data, or exploring advanced mathematics, this comprehensive logarithm calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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