Log Calculator
Calculate logarithms with any base including natural log (ln), common log (log10), and custom bases. Perfect for algebra, calculus, and scientific calculations with step-by-step explanations.
Logarithm Calculator
Common Logarithm Calculations
Click on these links to see instant calculations with common logarithms:
Logarithm Calculations
A logarithm is the inverse operation of exponentiation. The logarithm log_b(x) = y means that b^y = x, where b is the base, x is the argument, and y is the result.
Logarithm Types and Formulas
Natural Log: ln(x) or log_e(x) - base e ≈ 2.718
Binary Log: log₂(x) - base 2
Change of Base: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Common Logarithm Values
| Expression | Base | Number | Result |
|---|---|---|---|
| log₁₀(100) | 10 | 100 | 2 |
| ln(e) | e | 2.718 | 1 |
| log₂(8) | 2 | 8 | 3 |
| log₁₀(1000) | 10 | 1000 | 3 |
| log₂(16) | 2 | 16 | 4 |
- Mathematics: Solve exponential equations and work with exponential functions
- Science: Calculate pH levels, decibel measurements, and scientific scales
- Engineering: Analyze signal processing, control systems, and logarithmic scales
- Finance: Calculate compound interest, growth rates, and investment returns
- Computer Science: Algorithm analysis, data structures, and complexity calculations
Frequently Asked Questions
What is a logarithm?
A logarithm is the inverse operation of exponentiation. log_b(x) = y means that b^y = x. For example, log_2(8) = 3 because 2^3 = 8.
What is the difference between natural log and common log?
Natural log (ln) uses base e (≈2.718), while common log (log) uses base 10. ln(x) = log_e(x) and log(x) = log_10(x).
How do you calculate logarithms with different bases?
Use the change of base formula: log_b(x) = ln(x) / ln(b) or log_b(x) = log_10(x) / log_10(b). This converts any base to natural or common logarithms.