Exponent Calculator

An exponent calculator handles powers, roots, logarithms, and the natural exponential function in a single tool. Exponentiation is the operation of repeated multiplication: b^n means multiply b by itself n times. The base b is the number being multiplied, and the exponent n tells you how many times.

Roots, logarithms, and exponentials are all inverses of each other. The n-th root of x asks "what number raised to n gives x?" — equivalent to x^(1/n). A logarithm log_b(x) asks "what exponent on b gives x?" The natural exponential e^x and the natural logarithm ln(x) use the constant e ≈ 2.71828, which arises naturally in continuous growth, calculus, and probability.

A few worked examples across the operations:

• Power: 2^10 = 1,024 (this is why 1 KB ≈ 10^3 bytes is close to 2^10).
• Negative exponent: 5^-2 = 1 / 5^2 = 1/25 = 0.04.
• Cube root: 3√125 = 125^(1/3) = 5.
• Log base 10: log_10(1,000,000) = 6.
• Natural log: ln(7.389) ≈ 2 (because e^2 ≈ 7.389).
• Exponential: e^1 ≈ 2.71828, e^5 ≈ 148.41.

For continuous compound interest, $1,000 at 5% for 10 years gives 1000 × e^(0.05 × 10) = 1000 × e^0.5 ≈ $1,648.72.

• Compound interest and growth: populations, investments, bacteria — anything growing by a constant percentage uses exponentials.
• Scientific notation: express very large or small numbers like 6.022 × 10^23 (Avogadro's number) or 1.6 × 10^-19 C.
• Decibels and pH: these scales are logarithmic, so a 3-unit change is a thousand-fold change in the underlying quantity.
• Algorithm analysis: O(log n), O(n^2), and O(2^n) all show up in computer-science complexity.
• Geometry & physics: areas scale as length squared, volumes as length cubed, gravitational force as 1/r^2.

• Confusing -3^2 with (-3)^2. Without parentheses, the exponent binds tighter: -3^2 = -(3^2) = -9, while (-3)^2 = 9.
• Adding bases when multiplying: 2^3 × 3^3 ≠ 5^3. The product rule applies only to the same base.
• Treating log(a + b) as log(a) + log(b). The correct identity is log(a × b) = log(a) + log(b).
• Forgetting domain restrictions: log_b(x) requires x > 0 and b > 0, b ≠ 1. Negative or zero arguments are undefined in real numbers.
• Floating-point precision. Computed values like 0.1^3 may show as 0.0010000000000000002 due to binary rounding; round to the precision your problem actually needs.