Logarithm Calculator - Log, Ln, pH, Decibels, Graphing & More | 8gwifi.org

Logarithm Calculator

Calculate logarithms: log, ln, change of base, antilog with step-by-step solutions

Logarithm Calculator
Calculate: logbase(value)
Value must be positive
Examples:
• log₁₀(100) = 2
• ln(e) = 1
• log₂(8) = 3
Calculate antilog: basevalue
Examples:
• 10² = 100
• e¹ = 2.71828...
• 2³ = 8
Apply logarithm properties
Solve for x: logbase(x) = value
Examples:
• log₁₀(x) = 2 → x = 100
• log₂(x) = 3 → x = 8
• ln(x) = 1 → x = e
Simplify compound logarithmic expressions
Examples:
• log(8) + log(2) = log(16)
• log(100) - log(10) = log(10)
• log(x²y³) = 2·log(x) + 3·log(y)
Solve logarithmic inequalities
Examples:
• log(x) > 2 → x > 100
• log₂(x) < 3 → 0 < x < 8
• ln(x) ≥ 1 → x ≥ e
Real-world logarithm applications
pH = -log[H⁺]
Common Values:
• pH: Battery acid ≈ 0, Neutral ≈ 7, Bleach ≈ 13
• Decibels: Whisper ≈ 30 dB, Normal talk ≈ 60 dB
• Richter: Minor ≈ 3, Moderate ≈ 5, Major ≈ 7
Compare logarithms with different bases
Visualize logarithm curves
Key Features:
• Domain: x > 0 (logarithm undefined for x ≤ 0)
• Passes through (1, 0) for all bases
• Increases slowly as x increases
• Vertical asymptote at x = 0
Understanding Logarithms
What is a Logarithm?

A logarithm answers the question: "What power do I raise the base to, to get this number?"

If by = x, then logb(x) = y

Example: Since 2³ = 8, we say log₂(8) = 3

Common Logarithms

1. Common Logarithm (log):

log(x) = log₁₀(x)

Base 10. Used in science, engineering, pH scale, decibels

2. Natural Logarithm (ln):

ln(x) = loge(x)
where e ≈ 2.71828...

Base e (Euler's number). Used in calculus, exponential growth/decay

Logarithm Properties

Product Rule:

logb(xy) = logb(x) + logb(y)

The log of a product equals the sum of the logs

Quotient Rule:

logb(x/y) = logb(x) - logb(y)

The log of a quotient equals the difference of the logs

Power Rule:

logb(xn) = n · logb(x)

The log of a power equals the exponent times the log

Change of Base Formula:

logb(x) = logc(x) / logc(b)

Convert any base to base 10 or base e

Special Values
  • logb(1) = 0 (any base to power 0 equals 1)
  • logb(b) = 1 (base to power 1 equals itself)
  • logb(bn) = n (simplifies directly)
  • blogb(x) = x (log and exponent cancel)
Real-World Applications
  • pH Scale: pH = -log[H⁺] (acidity)
  • Richter Scale: Earthquake magnitude
  • Decibels: Sound intensity (dB = 10·log(I/I₀))
  • Half-life: Radioactive decay calculations
  • Compound Interest: Time to reach target amount
  • Population Growth: Exponential models
Important Notes
  • Cannot take log of negative numbers (in real numbers)
  • Cannot take log of zero
  • Base must be positive and ≠ 1
  • log(x) is undefined for x ≤ 0
Result

Enter values and click calculate to see logarithm result

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