Logarithm is defined only for x > 0 and base b > 0, b ≠ 1.
Step-by-step calculation example
x = 100: log₁₀(100) = 2, ln(100) ≈ 4.605, log₂(100) ≈ 6.644. Custom base 5: log₅(100) = ln(100)/ln(5) ≈ 2.861.
Key insights to remember
log₁₀(1000) = 3 because 10³ = 1000 — the common log counts the number of digits minus 1.
ln(e) = 1 where e ≈ 2.71828 (Euler's number). The natural log is the inverse of exponential growth.
log₂ is fundamental in information theory — log₂(1024) = 10 bits.
Change of base: log_b(x) = ln(x) / ln(b) — any logarithm can be computed from the natural log.
How to interpret your results?
Common log (log₁₀)
Used in pH (chemistry), Richter magnitude (earthquakes), and decibel (sound). log₁₀(x) answers '10 to what power equals x?'
Natural log (ln)
Used in calculus, probability, and finance (continuous compounding). It's the inverse of eˣ.
Binary log (log₂)
Fundamental in computer science: the number of bits needed to represent n values is ⌈log₂(n)⌉.
Frequently asked questions
How does the Log Calculator calculator work?
It uses standard algebraic, geometric, or statistical theorems to return exact values.
How accurate are the results?
All calculations use double-precision floating-point arithmetic for exact accuracy.
Can I copy the step-by-step work?
Yes, step-by-step solutions are formatted clearly for homework or documentation.
How are undefined operations handled?
The calculator flags mathematical errors such as division by zero or negative square roots.
Is this tool suitable for students and engineers?
Yes, it is designed for students, teachers, engineers, and scientists.
Does it support negative numbers and decimals?
Yes, all real numbers (positive, negative, fractions, and decimals) are supported.
Related calculators
Exponent Calculator — Calculate bᵉ for any base and exponent — positive, negative, or fractional — with scientific notation and exponent rules.
Scientific Notation Calculator — Convert any number to scientific notation and perform arithmetic (add, subtract, multiply, divide) on numbers in scientific notation.