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
log_b(x)=y exactly when b^y=x.
See the structure
What the calculation is doing
2³ = 8log_b(x)=y exactly when b^y=x.
Worked interpretation
Read the result in context
log₂(32)=5.
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 logarithm
Complex logarithms extend logs to negative and non-real inputs. Branch k changes the angle by 2πk.
Try an example
COMPLEX RESULT3 + 4.53236014183i
Principal branch
Understand the subject
What is a logarithm?
A logarithm answers an exponent question: what power of the base produces the given positive value?
Use logarithms to undo exponential relationships, compare orders of magnitude, and turn multiplication into addition.
The relationship
The defining equation
log_b(x)=y ⇔ bʸ=x
The input x must be positive; the base b must be positive and cannot equal 1.
See the structure
Read an exponential statement backward
2¹=22²=42³=82⁴=162⁵=32log₂(32)=5 counts the power
Worked example
Evaluate log₂(32)
Because 2⁵=32, log₂(32)=5
Interpret with care
What the result does—and does not—mean
A logarithm is not defined for zero or negative real inputs. Changing the base changes the numerical result but not the underlying exponential relationship.
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
log_b(x)=y exactly when b^y=x.
The calculator checks domains, zero denominators, unsupported powers, branches, and finite-range limits before reporting a result.
Worked example
Worked example
log₂(32)=5.
log_b(x)=y exactly when b^y=x.
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.
