How to use the exponent, root and logarithm calculator
- Choose an operation: power x^y, nth root, logarithm with any base, natural log or e^x.
- Enter the numbers asked for. Fields you do not need for the chosen operation are hidden.
- Read the result. Very large and very small results also appear in scientific notation.
- When both the base and the exponent of a power are whole numbers, check the exact value, which is computed digit by digit with big integers.
Formula
x^y, x^(-n) = 1 / x^n, x^0 = 1
nth root of x = x^(1/n)
log_b(x) = ln(x) / ln(b)
ln(x) = log base e, e = 2.718281828...
e^x, the inverse of ln(x)
A logarithm answers the question what power of b gives x. It is only defined for x > 0 and for a base b > 0 that is not 1.
Worked example
2^100 is exactly 1,267,650,600,228,229,401,496,703,205,376, a number with 31 digits, or about 1.2676506002 x 10^30. A normal double-precision calculation would round it, but whole-number powers use exact arithmetic here.
The fifth root of 243 is 3, because 3^5 = 243. Base 10 log of 1000 is 3, base 2 log of 1024 is 10, and ln(10) is 2.30258509299.
Domain rules the calculator enforces
A negative base with a fractional exponent has no real result, so it is rejected with an explanation. 0 to a negative power would divide by zero. The root of a negative number is real only for an odd whole degree, such as the cube root of -27, which is -3.
A logarithm needs a positive number and a positive base other than 1, because 1 raised to any power is still 1. For e^x, values of x above about 709.78 overflow double precision and values below about -745 underflow to zero, so they are reported instead of silently shown as infinity or 0.
Scientific notation
Results above 10^15 or below 10^-6 are shown as a number between 1 and 10 times a power of 10, for example 1.2676506002 x 10^30. This keeps long results readable and makes the size of the answer obvious at a glance.
Precision limits
Ordinary results use 64-bit floating point numbers, which keep about 15 to 17 significant digits and cover roughly 5 x 10^-324 to 1.8 x 10^308. Whole-number powers use exact integers up to 10,000 digits. For results longer than 15 digits the page shows the first 8 digits, the last 10 digits and the digit count, which is enough to check 2^100 or count the digits of 2^2000. A displayed log result that is within 10^-12 of a whole number k is snapped to k only when b^k really equals x.
Assumptions and limits
- Only real-number results are produced. Complex results are reported as undefined.
- 0^0 is shown as 1, the convention for powers, though some branches of math leave it undefined.
- Decimal powers and roots are computed in double precision and displayed with up to 12 significant digits.
- Exact big-integer results are limited to bases up to 1,000,000,000 in size and results up to 10,000 digits.
Frequently asked questions
What is a negative exponent?
It means a reciprocal. For example 2^-3 is 1 / 2^3, which is 0.125.
Why can't I raise a negative number to a fractional power?
Roots of negative numbers are not real for even degrees, and fractional powers need a choice of branch. Use whole-number exponents, or the root mode for odd roots.
What is the difference between ln and log?
ln is the logarithm with base e. A plain log means base 10 on most calculators, which you get here by choosing the any-base mode and a base of 10.
How is log base 3 of 81 found?
With the change of base rule, ln(81) / ln(3), which gives 4. The tool shows both logarithms in the steps.
Why does my huge power show an exact value?
Whole-number powers are computed with big integers, so the digits shown are exact even when the number is far beyond what a double can hold.