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Exponent, Root & Logarithm Calculator

Calculate powers (exponents), square/cube roots, logarithms, and convert scientific notation. Complete with formulas and reference tables.

MATHEMATICS

The Power, Root, and Logarithm Calculator is a comprehensive math tool for computing exponent operations and their inverses.

Four modes: power (a to the n, including decimals and negatives), root (square root, cube root, and nth root), logarithm (log base 10, ln, log base 2, and custom base), and scientific notation conversion. Each calculation includes solution steps, reverse verification, and a reference table.

Power, Root & Logarithm Calculator

Calculate powers, nth roots, logarithms, and scientific notation conversion with step-by-step solutions.

Hitung nilai b^n. Mendukung eksponen negatif, desimal, dan hasil besar/kecil dalam notasi ilmiah.

Calculator information

How to use this calculator

  1. Choose an operation: Power, Root, Logarithm, or Scientific Notation.
  2. For Power: enter the base (a) and the exponent (n). Exponents can be negative (e.g., -2) or decimal (e.g., 0.5).
  3. For Root: enter the number and root index (2 for square root, 3 for cube root, n for nth root).
  4. For Logarithm: select the base (10, e, 2, or custom) then enter the argument (the number whose log you want; must be > 0).
  5. Click Compute to see the result along with the solution steps and a verification check.
  6. Use the power/log reference tables for quick validation of results.

Power, Root, and Logarithm Operations

Power: a^n = a x a x ... x a (n times). Root: nth_root(a) = a^(1/n). Logarithm: log_b(x) = y means b^y = x.
  • a = base number
  • n = exponent (integer, negative, or fractional)
  • b = logarithm base (10 for common log, e ~= 2.71828 for ln, 2 for binary log)
  • x = argument, must be positive

Key properties: a^(-n) = 1/a^n, a^(m/n) = nth_root(a^m), log_b(xy) = log_b(x) + log_b(y).

Worked example: Compute 2^10, the cube root of 125, and log10(1000)

Given:
  • Power: base 2, exponent 10
  • Root: number 125, index 3
  • Logarithm: base 10, argument 1000
Steps:
  1. Power: 2^10 = 2 x 2 x 2 x ... (10 times) = 1,024.
  2. Cube root: find the number that, raised to the 3rd power, equals 125. 5^3 = 125, so cube_root(125) = 5.
  3. Logarithm: log10(1000) = what power of 10 produces 1000? 10^3 = 1000, so the result is 3.

Result: Results: 2^10 = 1,024; cube_root(125) = 5; log10(1000) = 3.

Frequently asked questions

What is the difference between log and ln?
Log (or log10) is the base-10 logarithm, commonly used in science and engineering. Ln (natural log) is base e ~= 2.71828, widely used in calculus, statistics, and physics. Conversion: ln(x) = log(x) / log(e) or log(x) = ln(x) / ln(10).
How do I compute negative and fractional exponents?
A negative exponent is a reciprocal: a^(-n) = 1/a^n, so 2^(-3) = 1/8 = 0.125. A fractional exponent equals a root: a^(1/n) = nth_root(a), so 16^(1/2) = 4 and 8^(1/3) = 2. Combined forms like a^(m/n) equal nth_root(a^m).
Why is the logarithm of zero or a negative number undefined?
The logarithm log_b(x) asks for the power of b that produces x. No real exponent can make a positive b produce zero or a negative number, because b^y is always positive for a positive base. Therefore the logarithm's domain is restricted to positive numbers (x > 0).
What is scientific notation and when is it used?
Scientific notation writes a number as a x 10^n where 1 <= |a| < 10. For example 6,500,000 is written as 6.5 x 10^6. This notation makes very large numbers (mass of the Earth) or very small numbers (mass of an electron) easier to write and is commonly used in physics, chemistry, and astronomy.
Does the square root of a negative number exist?
Not in the real numbers, because no real number squared yields a negative number. In complex numbers, however, sqrt(-1) = i (the imaginary unit) is defined, so sqrt(-9) = 3i. For everyday use (geometry, finance), restrict to the real domain.

Last updated: May 11, 2026

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