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((A/B)^p)=p log_b(A)−p log_b(B).
See the structure
What the calculation is doing
2³ = 8log_b((A/B)^p)=p log_b(A)−p log_b(B).
Worked interpretation
Read the result in context
log₁₀((100/10)^2)=2log₁₀100−2log₁₀10.
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.
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((A/B)^p)=p log_b(A)−p log_b(B).
The calculator checks domains, zero denominators, unsupported powers, branches, and finite-range limits before reporting a result.
Worked example
Worked example
log₁₀((100/10)^2)=2log₁₀100−2log₁₀10.
log_b((A/B)^p)=p log_b(A)−p log_b(B).
Supported inputs
Precision and limits
Real-number scope
This calculator returns real results and reports inputs outside its stated real domain.
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.
