Log Calculator Online Free Tool
Logarithm Calculator
Logarithm Equation
logb(x) = y ⟺ by = x
Provide any 2 values to calculate the 3rd
Logarithm Calculator
Special inputs: Use "e" for Euler's number (≈2.71828)
Must be positive
Can be any real number
A logarithm answers the question: what exponent do you raise the base to in order to get this number? This calculator computes log base 10, natural log (base e), log base 2, and any custom base. Logarithms are fundamental in science, engineering, finance, and computer science.
Logarithm Basics
log_b(x) = y means b^y = x. The three most common logarithms are base 10 (common log, written log or log₁₀), base e (natural log, written ln), and base 2 (binary log, written log₂, used in computer science).
log₁₀(1000) = 3 because 10³ = 1000 ln(e²) = 2 because e² = e² log₂(8) = 3 because 2³ = 8 Change of base: log_b(x) = ln(x) / ln(b) = log(x) / log(b)
e (Euler's number) ≈ 2.71828. Used in continuous growth/decay models.
Logarithm Rules
Product: log(ab) = log(a) + log(b) Quotient: log(a/b) = log(a) - log(b) Power: log(aⁿ) = n × log(a) log_b(b) = 1 log_b(1) = 0
Frequently Asked Questions
What is a logarithm used for in real life?⌄
The Richter scale for earthquakes is logarithmic (each step is 10× the energy). Decibels (sound) are logarithmic. pH (acidity) is logarithmic. Compound interest uses logarithms (Rule of 72 is a log approximation). Computer search algorithms (binary search) run in O(log n) time. Human perception of loudness and brightness follows logarithmic patterns.
What is the difference between log and ln?⌄
log (without a base specified) commonly means log₁₀ in most practical contexts (engineering, chemistry). ln means the natural logarithm with base e ≈ 2.71828. In pure mathematics and some scientific fields, log can mean the natural logarithm. Context matters -- always check which base is intended when a specific base is not stated.
Can you take the log of a negative number?⌄
Not in real numbers. The argument of a logarithm must be positive. log(0) is undefined (approaches negative infinity). Logarithms of negative numbers exist in complex mathematics but not in standard real-number calculations.
How do I calculate log base 5 on a calculator that only has log and ln?⌄
Use the change of base formula: log₅(x) = log(x) / log(5) or ln(x) / ln(5). Example: log₅(125) = log(125) / log(5) = 2.097 / 0.699 = 3. Check: 5³ = 125. Correct.