Exponential Function

Introduction

1.The exponential function is a function of the form:

$$f(x) = a^x, \quad a \in \mathbb{R}^+, \quad a \neq 1$$

2.For any base $a > 0,$ the graph of the exponential function $y = a^x$ always intersects the y-axis at the point $(0, 1),$ because $a^0 = 1.$ It lies entirely above the x-axis, since $a^x > 0$ for all $x \in \mathbb{R},$ and never intersects the x-axis, so $a^x \neq 0$ for any real $x.$ The behavior of the function depends on the value of the base $a,$ and three cases are distinguished.

Laws of exponents

3.The rules that hold for integer powers extend to every real exponent, provided the base is positive. For all real numbers $x$ and $y$ and bases $a, b > 0$:

$$ \begin{align} a^{x+y} &= a^x a^y \\[6pt] (a^x)^y &= a^{xy} \\[6pt] (ab)^x &= a^x b^x \end{align} $$

4.The first identity is the functional equation of the exponential function, since it turns a sum in the exponent into a product of values. Together with $a^0 = 1,$ it fixes the value of a negative exponent. Setting $y = -x$ gives $a^x a^{-x} = a^0 = 1,$ so:

$$a^{-x} = \frac{1}{a^x}$$

5.A negative exponent inverts the corresponding positive power, which agrees with the fact that $a^x$ stays positive for every real $x.$

Properties for $a$ greater than one

6.When $a > 1,$ the exponential function $y = a^x$ is strictly increasing over $\mathbb{R}.$

8.The limits as $x$ approaches the extremes of the domain are:

$$ \begin{align} \lim_{x \to -\infty} a^x &= 0^+ \\[6pt] \lim_{x \to +\infty} a^x &= +\infty \end{align} $$

9.

When $a > 1,$ the exponential function grows without bound as $x \to +\infty$ and approaches zero from above as $x \to -\infty.$ Each unit increase in $x$ multiplies the value of the function by the constant factor $a.$

Properties for $a$ between zero and one

10.When $0 < a < 1,$ the exponential function $y = a^x$ is strictly decreasing over $\mathbb{R}.$

12.The limits as $x$ approaches the extremes of the domain are:

$$ \begin{align} \lim_{x \to -\infty} a^x &= +\infty \\[6pt] \lim_{x \to +\infty} a^x &= 0^+ \end{align} $$

13.

When $0 < a < 1,$ the exponential function decreases without bound as $x \to -\infty$ and approaches zero from above as $x \to +\infty.$ Each unit increase in $x$ multiplies the value of the function by the constant factor $a,$ which is less than one.

Properties for $a$ equal to one

14.When $a = 1,$ the exponential function reduces to the constant function $y = 1^x = 1,$ which is excluded from the standard definition. Its graph is a horizontal line at height $y = 1.$

Connection with the logarithmic function

16.The exponential function $y = a^x$ is the inverse of the logarithmic function $y = \log_a(x),$ provided $a > 0$ and $a \neq 1.$ This inverse relationship means:

$$a^{\log_a(x)} = x, \qquad \log_a(a^x) = x$$

17.When the base $a$ equals Euler's number $e \approx 2.71828,$ the function is the natural exponential function:

$$f(x) = e^x$$

Exponential growth and decay

18.The natural exponential function is characterized by a differential equation together with an initial condition. It is the unique function whose rate of change equals its value at every point and whose value at the origin is one:

$$f'(x) = f(x), \qquad f(0) = 1$$

19.More generally, a quantity that changes at a rate proportional to its current size satisfies $f'(t) = k f(t),$ and the solutions of this equation are exactly the functions:

$$f(t) = C e^{kt}$$

20.Here $k$ is the proportional rate of change, and $C = f(0)$ is the value at the initial time $t = 0.$ The sign of $k$ sets the behavior. When $k > 0$ the quantity grows, multiplying by a fixed factor over each fixed time span; when $k < 0$ it decays toward zero. A population that doubles over equal intervals and a radioactive sample whose mass drops by a fixed proportion in equal times both obey this law.

21.For a decaying quantity the half-life is the time $t_{1/2}$ that reduces it to half its value. It follows from $C e^{k t_{1/2}} = \tfrac{1}{2} C$ by taking the natural logarithm of both sides:

$$t_{1/2} = -\frac{\ln 2}{k}$$

22.The constant $C$ cancels, so the half-life does not depend on the initial amount. For a growing quantity the same computation gives the doubling time $\ln 2 / k.$

23.The same $e^{-\lambda t}$ decay describes the exponential distribution, which models the waiting time between events that occur independently at a constant rate $\lambda > 0.$

Generalized exponential functions

24.Three cases arise when the base or the exponent is replaced by a function of $x.$

Limit, derivative and integral

26.The remarkable limit associated with the natural exponential function is:

$$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$$

27.This limit shows that the derivative of $e^x$ at the origin equals one, consistently with $\frac{d}{dx} e^x = e^x.$ For a general base $a > 0,$ $a \neq 1,$ the corresponding limit is:

$$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$$

28.The derivative of the exponential function follows directly from the limit above. Differentiating $a^x$ with respect to $x$ gives:

$$\frac{d}{dx} a^x = a^x \ln(a)$$

$$\frac{d}{dx} e^x = e^x$$

29.Differentiating once more shows that the exponential function is convex for every admissible base. The second derivative is:

$$\frac{d^2}{dx^2} a^x = a^x (\ln a)^2$$

30.Since $a^x > 0$ and $(\ln a)^2 > 0$ whenever $a \neq 1,$ the second derivative stays positive for every $x \in \mathbb{R},$ so the graph always bends upward, both when the function increases and when it decreases.

31.The integral of the exponential function is obtained by reversing the differentiation formulas above:

$$\int a^x \ dx = \frac{a^x}{\ln(a)} + c$$

$$\int e^x \ dx = e^x + c$$

Asymptotic growth

32.The exponential function grows faster than any polynomial or power function, and slower than the factorial. For any $a > 1$ and any $k > 0$:

$$\lim_{x \to +\infty} \frac{x^k}{a^x} = 0 \qquad \lim_{x \to +\infty} \frac{a^x}{x!} = 0$$

33.This establishes the following hierarchy of growth rates as $x \to +\infty$:

$$\log x \ll x^k \ll a^x \ll x!$$

34.The table below illustrates this hierarchy for $a = 2.$

35.

The hierarchy $\log x \ll x^k \ll a^x \ll x!$ appears in asymptotic analysis and in the classification of algorithm complexity in computer science, where time and space requirements are measured against these growth rates.

Hyperbolic functions derived from the exponential function

36.Combining $e^x$ and $e^{-x}$ yields the hyperbolic functions, which appear in analysis and geometry. The three main ones are the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent, defined as follows:

$$\cosh(x) = \frac{e^{x} + e^{-x}}{2} \quad x \in \mathbb{R}$$

$$\sinh(x) = \frac{e^{x} - e^{-x}}{2} \quad x \in \mathbb{R}$$

$$\tanh(x) = \frac{\sinh(x)}{\cosh(x)} \quad x \in \mathbb{R} \quad \tanh(x) \in (-1, 1)$$

37.Because they are defined through the exponential function, hyperbolic functions are smooth and differentiable over $\mathbb{R}.$ The function $\tanh(x)$ is bounded, unlike $\sinh(x)$ and $\cosh(x),$ which grow without bound as $x \to \pm\infty.$

38.

Their definitions mirror those of the trigonometric functions, with the difference that $\cosh$ and $\sinh$ parametrize the unit hyperbola $x^2 - y^2 = 1$ rather than the unit circle $x^2 + y^2 = 1.$