Logarithm Calculator
Calculate the logarithm of any number with a custom base, natural logarithm (ln) and common logarithm (log10). Fast online logarithm calculator.
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Roots and powers calculator
This tool computes square roots, cube roots, nth roots and integer or fractional powers. Choose the operation, enter the base and — where needed — the exponent or root degree. Negative bases are supported for odd-degree roots and all exponents in power mode.
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How we calculate roots and powers
Square root: √x = x^(1/2). Cube root: ∛x = x^(1/3). Nth root: â¿âˆšx = x^(1/n). Power: aâ¿ = a multiplied by itself n times. For even-degree roots of negative numbers the result is undefined in real numbers.
Example calculation
Square root of 144: operation = sqrt, base = 144 → result = 12 (since 12² = 144). Power example: 2 raised to the 10th power = 2¹Ⱐ= 1024.
Frequently asked questions
What is a square root?
The square root of x is a number y such that y² = x. For example √9 = 3. It is defined only for non-negative real numbers.
How do you calculate a cube root?
The cube root of x is y where y³ = x. Example: ∛27 = 3. Unlike the square root, cube roots exist for negative numbers: ∛(−8) = −2.
What is an nth root?
The nth root of x is y satisfying y⿠= x. For even n, x must be ≥ 0 in the reals; for odd n, negative x is allowed.
How does exponentiation work?
aâ¿ means a multiplied by itself n times. Example: 2âµ = 32. Any non-zero number raised to the 0th power equals 1.
What is a negative exponent?
aâ»â¿ = 1/aâ¿. Example: 2â»Â³ = 1/8 = 0.125. Negative exponents represent reciprocals.
Can you take the square root of a negative number?
Not in real numbers. The square root of a negative number is imaginary (involves i). This calculator works in real numbers only.
What is the relationship between roots and fractional exponents?
The nth root of x equals x^(1/n). Example: √9 = 9^(0.5) = 3, ∛8 = 8^(1/3) = 2.
How do you calculate the root of a fraction?
√(p/q) = √p / √q. Example: √(4/9) = 2/3. Enter 0.4444 as the base to get the same result.
What is a fractional power?
x^(m/n) = (â¿âˆšx)^m. Example: 8^(2/3) = (∛8)² = 2² = 4. Fractional exponents combine roots and powers.
Why does aâ° = 1?
From the quotient rule: a⿠/ a⿠= a^(n−n) = aⰠ= 1. This holds for all a ≠0. The expression 0Ⱐis indeterminate.
Results are indicative. The calculator operates on real numbers — even-degree roots of negative values return an error.