Calculator inputs
Use Cartesian or polar form, choose a branch, and optionally divide by a complex base logarithm.
Example data table
| z | Input form | k | Principal ln(z) | Selected result |
|---|---|---|---|---|
| 3 + 4i | Cartesian | 0 | 1.609438 + 0.927295i | 1.609438 + 0.927295i |
| 3 + 4i | Cartesian | 1 | 1.609438 + 0.927295i | 1.609438 + 7.210481i |
| 5∠53.130102° | Polar | -1 | 1.609438 + 0.927295i | 1.609438 - 5.355890i |
| -2 + 2i | Cartesian | 0 | 1.039721 + 2.356194i | log10(z) = 0.451545 + 1.023283i |
Formula used
1) Complex logarithm
Logk(z) = ln|z| + i(Arg(z) + 2πk), where k is any integer and z ≠ 0.
2) Principal value
The principal logarithm uses k = 0. Its argument follows the selected interval: (-π, π] or [0, 2π).
3) Polar connection
If z = r(cos θ + i sin θ), then |z| = r and the logarithm becomes ln r + i(θ + 2πk).
4) Base conversion
logb(z) = Logk(z) / Logj(b). A valid base requires b ≠ 0 and the chosen Logj(b) ≠ 0.
How to use this calculator
- Choose Cartesian form for real and imaginary inputs, or polar form for magnitude and angle.
- Select degrees or radians, then choose the principal argument range that matches your convention.
- Pick natural, common, or custom-base logarithm. For a custom base, enter its real and imaginary parts.
- Enter branch index
k. For custom bases, enter base branchjas well. - Set the output precision and press Submit.
- The result panel above the form shows the principal value, selected branch, and base-converted answer. Use the buttons there for CSV or PDF exports.
FAQs
1) Why does the complex logarithm have many values?
Angles repeat every 2π. Adding 2πk changes the imaginary part but keeps the modulus term unchanged, so every integer k creates another valid logarithm branch.
2) What is the principal complex logarithm?
It is the branch with k = 0. The argument comes from the chosen principal interval, so the result is a single representative value among infinitely many branches.
3) When should I use Cartesian input?
Use Cartesian input when your number is already written as x + yi. It is convenient for algebraic work and helps you inspect the real and imaginary parts directly.
4) When should I use polar input?
Use polar input when magnitude and angle are known or when your problem is already expressed geometrically. It makes branch reasoning and argument adjustments easier to visualize.
5) Why is z = 0 not allowed?
Because ln|z| appears in the formula. At z = 0, the modulus is zero and ln(0) is undefined, so the complex logarithm does not exist there.
6) Can the base also be complex?
Yes. This page supports complex bases and a separate branch index j for the base logarithm. That is useful when studying multi-valued base-conversion behavior.
7) Why can two software tools show different answers?
They may use different principal argument ranges, different branches, or different base conventions. Matching those settings usually reconciles the answers.
8) What does changing branch k actually do?
It adds 2πk to the argument, so only the imaginary component shifts. The real component ln|z| stays the same for every branch.