Log Calculator
Calculate logarithms with various bases including natural and common logs. Free, fast, accurate — no signup, mobile-frie
Calculation Type
Number to find logarithm of
Base of logarithm (must be > 0 and ≠ 1)
📐 Common Logarithms
Logarithm Properties
Main Result
log₍10₎(100)
Natural Log
ln(x) = log₍ₑ₎(x)
Common Log
log₁₀(x)
Binary Log
log₂(x)
Different Notations
Function Properties
Step-by-Step Solution
How it works
A logarithm answers “what power do I raise the base to, to get this number?” It's the inverse of exponentiation. Common bases are 10 (log) and e (natural log, ln). Logs turn multiplication into addition, which is why they appear in scales like decibels and pH.
Logarithm
logᵦ(x) = y means bʸ = x
- b
- the base
- x
- the number
- y
- the exponent (the log)
Worked example
- Compute log₁₀(1000)
- What power of 10 gives 1000?
- 10³ = 1000
log₁₀(1000) = 3.
Good to know
- log(ab) = log a + log b — turning products into sums is the whole point of logarithms.
- Change of base: logᵦ(x) = ln(x) ÷ ln(b), so any calculator with ln can do any base.
- You can't take the log of zero or a negative number.
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Frequently Asked Questions
What is a logarithm?
A logarithm is the inverse of exponentiation: log_b(x) = y means bʸ = x. So log₂(8) = 3 because 2³ = 8. Logarithms answer the question "what exponent do I need?"
What's the difference between log and ln?
log usually means the common logarithm (base 10), while ln is the natural logarithm (base e ≈ 2.71828). Base 10 suits orders of magnitude (pH, decibels, Richter scale); base e arises naturally in calculus, growth, and decay problems.
How do I calculate a log with an unusual base?
Use the change-of-base formula: log_b(x) = ln(x) ÷ ln(b) (any common base works). For example, log₇(50) = ln 50 ÷ ln 7 ≈ 2.011. This is how calculators evaluate arbitrary bases internally.
What are the basic logarithm rules?
Product: log(xy) = log x + log y. Quotient: log(x/y) = log x − log y. Power: log(xⁿ) = n·log x. These rules turn multiplication into addition and powers into multiplication, which is the historical reason logs were invented.
Why can't I take the log of zero or a negative number?
Because no real exponent makes a positive base produce zero or a negative result — bʸ is always positive. Logarithms of non-positive numbers are undefined in the real numbers (complex analysis extends them, but standard calculators return an error).