Calculate log with any base | Natural log (ln) | Common log (lg) | Log base 2
Result
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📝 Step-by-step explanation
Enter values and click calculate.
📖 How to Use This Logarithm Calculator
1
Select logarithm type - Choose Custom Base, Natural Log (ln), Common Log (lg), or Log Base 2.
2
Enter the base (for custom log) - Input the base of the logarithm (must be >0 and ≠1).
3
Enter the number (x) - Input the number you want to find the logarithm of (must be >0).
4
Click "Calculate" - Get the result with step-by-step explanation.
📐 What is a Logarithm?
A logarithm answers the question: "To what exponent must the base be raised to get a given number?"
The logarithm of x with base b is written as logₐ(x) = y, which means aʸ = x. For example, log₁₀(100) = 2 because 10² = 100.
📊 Common Logarithm Types
Common Logarithm (lg / log₁₀): Base 10. Used in scientific notation, decibels, pH scale.
Natural Logarithm (ln): Base e (≈ 2.71828). Used in calculus, exponential growth/decay, compound interest.
Binary Logarithm (lb / log₂): Base 2. Used in computer science, information theory, data structures.
Custom Base: Any positive number ≠ 1. Used in specialized applications.
🧮 Logarithm Rules and Properties
Product Rule: logₐ(m × n) = logₐ(m) + logₐ(n)
Quotient Rule: logₐ(m / n) = logₐ(m) - logₐ(n)
Power Rule: logₐ(mⁿ) = n × logₐ(m)
Change of Base: logₐ(b) = log_c(b) / log_c(a)
Special Values: logₐ(1) = 0, logₐ(a) = 1
💡 Real-World Applications of Logarithms
Richter Scale (Earthquakes): Logarithmic scale where each whole number increase represents 10x amplitude
Decibels (Sound): Logarithmic scale for sound intensity and power
pH Scale: Measures acidity as negative logarithm of hydrogen ion concentration
Compound Interest: Calculating time to reach investment goals
Algorithm Complexity: Binary search, merge sort use log₂ complexity
❓ Frequently Asked Questions
What is the difference between log and ln?
Log typically means log₁₀ (common logarithm) while ln means natural logarithm (base e ≈ 2.71828). Otherwise, they follow the same rules.
Why can't a logarithm have a negative base?
Logarithms with negative bases are not defined in real numbers because they produce complex results. For real-valued logarithms, the base must be positive and not equal to 1.
What is the log of 0?
logₐ(0) is undefined (approaches -∞). There's no exponent that gives zero because a positive base raised to any power is always positive.