Powers, reciprocals and roots
A positive integer exponent means repeated multiplication: 2¹⁰ = 1,024. A negative exponent gives a reciprocal, so 2⁻³ = 1 ÷ 8 = 0.125. For positive bases, a fractional exponent describes a root: 9^0.5 = 3.
To solve b^e = r for the exponent, use e = ln(r) ÷ ln(b), requiring b > 0, b ≠ 1 and r > 0. To find a principal base for a non-zero exponent, use r^(1/e), subject to the real-number domain.
Example: recover the base and exponent
For b³ = 125, the base is 125^(1/3) = 5. For 2^e = 1,024, e = ln(1,024) ÷ ln(2) = 10. An even integer exponent may have two real bases: b² = 25 has b = +5 and b = −5. The calculator shows both signs and uses the positive base for its reference table.
Real-number and precision limits
Negative bases with integer exponents work, including odd roots of negative results in base-solving mode. Decimal exponents on negative bases are unsupported; a rounded decimal does not reliably identify an exact rational root.
Zero to a positive exponent is zero. This tool treats 0^0 and zero to a negative exponent as undefined, and cannot recover a unique base from exponent zero. The logarithm mode does not solve negative-base equations. Overflow is rejected; very small powers may underflow to zero. Results use floating-point arithmetic rather than symbolic exact algebra.