How this calculator works
Positive bases support real exponents. Negative bases are evaluated only for integer exponents here. Recovering an exponent uses a quotient of logarithms; base 1 cannot determine one. Recovering a base gives the positive branch for positive values and also reports the negative branch for even integer powers. Negative known values are supported only with odd integer exponents. Extremely large results or underflow are rejected.
Formula
y=b^x; x=ln(y)/ln(b) for b>0, b≠1, y>0; b=y^(1/x) on the positive branch for x≠0.
Worked example
2^5=32. Reversing 3^x=81 gives x=ln(81)/ln(3)=4. Reversing b^4=16 gives b=2 or b=-2, while b^3=-8 gives b=-2.
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Frequently asked questions
Why are there sometimes two bases?
An even integer exponent gives the same positive value for equal positive and negative bases.
Can a decimal exponent use a negative base?
Not in this tool. Use the fractional exponent calculator for an exact rational exponent with an odd reduced denominator.
Why exclude 0^0?
This calculator leaves that indeterminate convention undefined. Zero is accepted only with a positive exponent.