e^x Calculator - Calculate e to the Power of x

What Is $e^x$ (Natural Exponential Function)?

The natural exponential function, written as $e^x$ or $\exp(x)$, is the exponential function whose base is the mathematical constant $e$ (Euler's number), approximately equal to $2.718281828459$. It plays a foundational role in calculus, differential equations, physics, finance, and probability.

Mathematical Definition & Euler's Constant

Euler's number $e$ can be defined as the limit of compound growth as compounding intervals approach infinity:

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828459\dots$$

The natural exponential function $f(x) = e^x$ is the unique nonzero function whose derivative is equal to itself:

$$\frac{d}{dx} e^x = e^x, \quad \int e^x dx = e^x + C$$

Taylor / Maclaurin Series Expansion

For any real or complex value of $x$, $e^x$ can be evaluated via its infinite Taylor (Maclaurin) series expansion, which converges for all $x \in \mathbb{R}$:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \dots$$

This rapid convergence allows high-precision numerical computation for positive, negative, and fractional exponents.

Key Properties of $e^x$

Property	Algebraic Formula	Notes
Zero Exponent	$e^0 = 1$	Passes through the point $(0, 1)$
Negative Exponent	$e^{-x} = \frac{1}{e^x}$	Reciprocal exponential decay
Product Rule	$e^a \cdot e^b = e^{a+b}$	Adding exponents
Quotient Rule	$\frac{e^a}{e^b} = e^{a-b}$	Subtracting exponents
Power of a Power	$(e^a)^b = e^{a \cdot b}$	Multiplying powers
Inverse Relation	$\ln(e^x) = x \quad \text{and} \quad e^{\ln(y)} = y \ (y > 0)$	Natural log is the inverse of $e^x$

Applications in Science and Finance

Continuous Compounding Interest: Future value $A = P e^{rt}$, where $P$ is principal, $r$ is annual rate, and $t$ is time in years.
Population Growth and Radioactive Decay: Modeled by $N(t) = N_0 e^{kt}$ ($k > 0$ for growth, $k < 0$ for decay).
Hyperbolic Trigonometry: Defined as $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
Euler's Identity: $e^{i\pi} + 1 = 0$, uniting $e$, $i$, $\pi$, $1$, and $0$.

Related Calculators

Natural Log Calculator: Calculate natural logarithms $\ln(x)$ and logarithmic properties.
Log Calculator: Compute logarithms to any custom base.
Power Calculator: Evaluate any general exponent $a^b$.
Logarithmic Growth Calculator: Model and analyze exponential and logarithmic growth trends.

Frequently Asked Questions

What is the value of e?

The value of Euler's constant $e$ is an irrational, transcendental number approximately equal to $2.718281828459045$.

Can e^x ever be negative or zero?

No. For any real number $x$, $e^x$ is strictly positive ($e^x > 0$). As $x \to -\infty$, $e^x$ approaches $0$ asymptotically, but it never reaches zero or becomes negative.

How do you solve for x in e^x = y?

Take the natural logarithm ($\ln$) of both sides: $\ln(e^x) = \ln(y) \implies x = \ln(y)$. This requires $y > 0$.

What is the difference between e^x and 10^x?

$e^x$ uses base $e \approx 2.718$ (the natural base, where the slope of the curve at any point equals its y-value), while $10^x$ uses base $10$ (the common decimal base used in powers of ten and decibels).

What is the derivative of e^(kx)?

Applying the chain rule, $\frac{d}{dx} e^{kx} = k \cdot e^{kx}$.

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