Exponent Calculator

The exponential function e^x uses the mathematical constant e (≈ 2.71828) as its base. It describes continuous growth and decay processes — population growth, radioactive decay, compound interest, and thermal cooling all follow exponential patterns.

Key properties of e^x:

Inverse of natural logarithm: If e^x = y, then ln(y) = x. These two functions undo each other.

Derivative equals itself: d/dx(e^x) = e^x. The exponential function is its own derivative — unique among all functions.

Power properties: e^a x e^b = e^a+b. (e^a)^b = e^ab.

e⁰ = 1: Any number raised to the zero power equals 1.

Negative exponents: e^-x = 1/e^x.

Exponential Rules

• e⁰ = 1
• e^a x e^b = e^a+b
• e^a / e^b = e^a-b
• (e^a)^b = e^ab
• e^-a = 1/e^a

Where Exponential Functions Appear

Finance: Compound interest, continuous compounding, option pricing models.

Biology: Population growth models, bacterial colony expansion, drug half-life in the bloodstream.

Physics: Radioactive decay, capacitor charge/discharge, Newton's law of cooling.

Engineering: Signal processing, control systems, damping in mechanical systems.

Computer Science: Machine learning activation functions, cryptographic algorithms, complexity analysis.