Log calculator for any real base, natural log and common log
Enter a positive value and valid base. HitEquals returns log base b, ln, log base 10 and the inverse power check.
Inputs used by this calculator
Log Calculator uses the fields below. The starting values reproduce the page example; replace them with values from the same units and time period before using the result.
| Input | Starting example | Accepted value | How it is used |
|---|---|---|---|
| Positive value | 1,000 | 1e-300 to 1e+300 | Supplies the positive value quantity used by the published equation or result breakdown. |
| Logarithm base | 10 | 1e-12 to 1000000000000 | Supplies the logarithm base quantity used by the published equation or result breakdown. |
What a logarithm means
The equation log_b(x)=y says exactly the same thing as bʸ=x. Logarithms invert exponential functions.
Change of base
A calculator can evaluate any valid base by dividing two natural logarithms.
log_b(x)=ln(x)/ln(b)Common and natural logs
Common log uses base 10. Natural log uses base e and is written ln.
Logarithm rules
Products become sums, quotients become differences and powers become multipliers when all expressions remain in the real domain.
log_b(xy)=log_b(x)+log_b(y)log_b(x^k)=k log_b(x)Worked power example
Because 10³=1000, log₁₀(1000)=3. The inverse check raises 10 to the computed result.
Domain and numerical limits
Zero and negative real arguments have no real logarithm, and base one cannot produce values other than one. Bases between zero and one are valid and yield a decreasing logarithmic function.
Common logarithm bases
| Name | Base | Notation |
|---|---|---|
| Common | 10 | log(x) |
| Natural | e | ln(x) |
| Binary | 2 | log₂(x) |
| General | b | log_b(x) |
Frequently asked questions
What is log base 10 of 1000?
It is 3.
What is a natural logarithm?
It is a logarithm with base e.
Can the value be zero?
No. A real logarithm requires a positive argument.
Can the base be one?
No. Powers of one cannot create an invertible exponential function.
Can the base be between zero and one?
Yes. It produces a decreasing logarithmic function.
How does change of base work?
Divide ln(value) by ln(base).
What to keep in mind
Real logarithms require a positive argument and positive base other than 1. This tool does not calculate complex logarithms.
Results are rounded for display in US format, with money in dollars and cents; CSV exports keep full calculation precision. This is a calculation summary, not an official certificate.