Log Base N Calculator
Calculate logarithms with any custom base, including step-by-step solutions
Expression:
log() =
log()
=
Step-by-Step Solution
Common Log (log₁₀)
log₁₀()
Natural Log (ln)
ln()
Binary Log (log₂)
log₂()
Logarithm Properties Reference
| Property | Formula |
|---|---|
Common Logarithm Bases
Find this log base n calculator helpful?
Please help us simply by sharing it. It will help us a lot!
Related Calculators
Other calculators you might find useful.
Significant Figures Calculator
Count sig figs, round to significant figures, and perform calculations with proper precision
Variance Calculator
Calculate population and sample variance with detailed statistics
Binomial Probability Distribution Calculator
Calculate probability of k successes in n independent trials with step-by-step solutions
Midpoint Calculator
Calculate coordinates of the midpoint between two points in 2D or 3D space
Proportion Calculator
Solve proportions using cross-multiplication, find missing values, and verify if ratios are proportional
Percentage Decrease Calculator
Calculate the percentage decrease between two values with step-by-step solutions
About Log Base N Calculator
What is Log Base N?
Log Base N (written as log_n(x)) is the logarithm of a number x to base n. It answers the question: "To what power must n be raised to get x?" If log_n(x) = y, then n^y = x.
How to Use This Calculator
- Enter the Number (x): Input the positive number you want to find the logarithm of
- Enter the Base (n): Input any positive base (except 1)
- View Instant Results: See the result with step-by-step explanation
- Compare Bases: View the same number's logarithm in common bases (2, e, 10)
The Formula
Log Base N Formula: log_n(x) = ln(x) / ln(n) = log₁₀(x) / log₁₀(n)
This is known as the Change of Base Formula, which allows calculating logarithms of any base using natural or common logarithms.
Common Logarithm Bases
Binary Logarithm (Base 2)
- Used in computer science, information theory
- log₂(8) = 3 because 2³ = 8
Natural Logarithm (Base e ≈ 2.71828)
- Used in calculus, physics, engineering
- ln(e) = 1 because e¹ = e
Common Logarithm (Base 10)
- Used in scientific notation, pH scale, decibels
- log₁₀(100) = 2 because 10² = 100
Logarithm Properties
Product Rule
log_n(ab) = log_n(a) + log_n(b)
Quotient Rule
log_n(a/b) = log_n(a) - log_n(b)
Power Rule
log_n(a^k) = k × log_n(a)
Change of Base
log_a(x) = log_b(x) / log_b(a)
Frequently Asked Questions
What is a valid base for logarithms?
The base must be positive and not equal to 1. Common bases include 2, e (≈2.718), 10, and any other positive number.
Why can't the base be 1?
If the base were 1, then 1^y = 1 for any y, meaning 1^y can never equal any number other than 1. This makes the logarithm undefined for x ≠ 1.
Why can't x be negative or zero?
Logarithms are only defined for positive real numbers. log_n(0) approaches negative infinity, and log_n(negative) requires complex numbers.
How do I convert between logarithm bases?
Use the change of base formula: log_a(x) = log_b(x) / log_b(a). This lets you calculate any logarithm using a calculator that only has ln or log₁₀.
Real-World Applications
- Earthquake Magnitude: The Richter scale uses log₁₀
- Sound Intensity: Decibels use log₁₀
- pH Scale: Measures acidity using -log₁₀
- Algorithm Complexity: O(log n) in computer science
- Compound Interest: Time calculations use natural log
Note: This calculator provides accurate results for any positive base (except 1) and any positive number x.