Exponents describe repeated multiplication, roots undo a power relationship, and logarithms name the exponent needed to reach a value.
Use it to move transparently between a base, exponent, root, logarithm, and equivalent notation.
The relationship
Write the rule before calculating
ln(x)=log_e(x), for x>0.
See the structure
What the calculation is doing
2³ = 8ln(x)=log_e(x), for x>0.
Worked interpretation
Read the result in context
ln(e)=1.
Interpret with care
Important boundary
Even roots require a non-negative real radicand here, while logarithms require a positive argument and valid base.
Use the Mathematics collection to move between connected concepts without duplicating the calculation.
Extended domain
Complex natural logarithm
The principal value is ln|z| + i arg(z); other branches add 2πki.
Try an example
COMPLEX RESULT3.14159265359i
Principal branch
Quick guide
How to use this calculator
Choose a calculation mode when the page offers more than one relationship.
Enter the labelled known values.
Read the primary result and use the check or equivalent form to verify its meaning.
Calculation method
Apply the exponent, root, or logarithm relationship
ln(x)=log_e(x), for x>0.
The calculator checks domains, zero denominators, unsupported powers, branches, and finite-range limits before reporting a result.
Worked example
Worked example
ln(e)=1.
ln(x)=log_e(x), for x>0.
Supported inputs
Precision and limits
Real and complex scope
The primary panel handles the familiar real case; the extended-domain panel accepts real and imaginary parts, identifies the selected logarithm branch, and plots complex results.
Numerical range
Inputs may be zero or have absolute magnitude from 1e-100 through 1e100. Exact integer radical simplification uses its narrower disclosed bound.
Precision
Numerical functions use double-precision arithmetic and normally display 12 significant digits. Symbolic labels are exact only where explicitly stated.
