Log Calculator Online
Calculate logarithms for any base, natural log ln(x), common log log10(x), and binary log log2(x).
Input Parameters
Live Result
Details
Steps
Identify Logarithmic Form: log_10(100) = y <=> 10^y = 100
Apply Change of Base Formula: log_b(x) = ln(x) / ln(b)
Evaluate Natural Logarithms: ln(100) ā 4.60517019, ln(10) ā 2.30258509
Divide Values: 4.605170 / 2.302585 = 2
Exponential Verification: 10^(2) ā 100.0000
Quick Answer & Understanding Your Result
A Logarithm Calculator determines the power to which a base must be raised to produce a given number: log_b(x) = y if and only if b^y = x. The calculation requires the argument to be strictly positive (x > 0) and the base to be positive and non-unit (b > 0, b ā 1).
The 3 Core Standard Logarithm Bases
- Natural Logarithm (ln x): Base is Euler's number e ā 2.71828, fundamental in physics and calculus.
- Common Logarithm (log10 x): Base 10, foundational for pH, Richter scale earthquakes, and decibels.
- Binary Logarithm (log2 x): Base 2, essential for computer science algorithms and information entropy.
- Change of Base Formula: log_b(x) = ln(x) / ln(b) = log10(x) / log10(b).
Fundamental Laws of Logarithms
- Product Rule: log_b(m Ć n) = log_b(m) + log_b(n).
- Quotient Rule: log_b(m / n) = log_b(m) - log_b(n).
- Power Rule: log_b(m^p) = p Ć log_b(m).
- Identity Rules: log_b(b) = 1, and log_b(1) = 0.
Enter your argument number and base into the inputs above to compute exact logarithms and exponential verification.
Formula & Calculation Logic
Mathematical FormulaThe logarithm is the inverse operation to exponentiation. It answers: To what power must the base b be raised to equal x?
Step-by-Step Guide
Enter Argument (x)
Input argument value x (must be strictly greater than 0).
Enter Base (b)
Input base b (positive number ā 1, or enter 'e' for natural log).
Choose Decimal Precision
Select desired decimal places (from 4 up to 8 decimals).
Review Output
View calculated log_b(x), natural log, common log, and exponential verification.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Common Log log10(100) | 10Ā² = 100, so log10(100) = 2. | ||
| Binary Log log2(1024) | 2Ā¹ā° = 1024, so log2(1024) = 10. | ||
| Natural Log ln(eĀ²) | eĀ² ā 7.389056, so ln(eĀ²) = 2. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
Wolfram MathWorld - Logarithm
NIST Digital Library of Mathematical Functions - Logarithms
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: log_b(x) = y <ā b^y = x
Last Reviewed: August 21, 2026
Primary Source: Wolfram MathWorld - Logarithm
Reference Document: NIST Digital Library of Mathematical Functions - Logarithms
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.