Exponential function: Definition & Examples

An exponential function has a fixed positive base and a variable in the exponent:

f(x)=a^x,\qquad a>0,\quad a\ne1.

The input $x$ can be any real number, and the output is always positive. The base restriction matters: a negative base does not give a real-valued function for every real input, while zero fails for negative inputs. The excluded case $a=1$ produces the constant function $1$ .

The location of the variable distinguishes the family. In

2^x

, the base stays fixed while the exponent changes. In

x^2

, the exponent stays fixed while the base changes: that is a power function. Their algebra and differentiation rules differ.

Equal steps, equal multipliers

The defining operational property is

f(x+h)=a^{x+h}=a^x a^h=f(x)a^h.

A fixed step

h

in the input multiplies the output by the same factor

a^h

, wherever the step starts. It does not add a fixed amount. When

a>1

, increasing the input produces exponential growth; when

0<a<1

, it produces exponential decay. Both curves pass through

(0,1)

and remain above zero.

Multiplying by a positive starting value $C$ gives $y(x)=Ca^x$ , a scaled exponential. Its equal-step ratio is still $a^h$ , but $y(0)=C$ . Be careful with the identity $f(x+h)=f(x)f(h)$ : it applies to the normalised function $a^x$ . For the scaled version, $y(x+h)=y(x)y(h)/C$ .

A cash balance with an exponential clock

Suppose a hypothetical account starts with USD 1,000, earns a stipulated constant 5% continuously compounded annual rate, and has no deposits, withdrawals, fees or taxes. With

t

measured in years, its model balance is

B(t)=1{,}000e^{0.05t}.

Here $e\approx2.71828$ is the natural base. The normalised accumulation factor $f(t)=e^{0.05t}$ starts at $1$ and satisfies $f(t+s)=f(t)f(s)$ . The exponent $0.05t$ is dimensionless: an annual rate multiplies a time in years.

Time	Model balance	Gain over preceding two years
0 years	USD 1,000.00	—
2 years	USD 1,105.17	USD 105.17
4 years	USD 1,221.40	USD 116.23

Each two-year interval multiplies the balance by $e^{0.1}\approx1.105171$ . The second dollar gain is larger because the same multiplier acts on a larger starting balance.

A stipulated USD 1,000 account at a constant 5% continuous annual rate reaches USD 1,105.17 after two years and USD 1,221.40 after four. Each two-year multiplier is 1.105171; the gains are USD 105.17 and USD 116.23.

Original hypothetical model: equal two-year intervals share one multiplier, while their dollar gains differ. Rate and time units are fixed throughout.

This is a specified accumulation rule, not a forecast of an investment's realised returns. A 5% continuous rate also differs from a 5% effective annual rate: the former produces an annual gain of $e^{0.05}-1\approx5.127\%$ . Changing rates or external cash flows require a different balance model.

Why the natural base appears in calculus

Every positive base can be written using a logarithm:

a^x=e^{x\ln a}

. Differentiating shows that an exponential's rate of change is proportional to its current level:

f'(x)=(\ln a)f(x), \qquad \frac{f'(x)}{f(x)}=\ln a.

For the account,

B'(t)=0.05B(t)

. At two years the instantaneous slope is about USD 55.26 per year. This is a local rate of change; the exact gain over the next year uses the multiplier

e^{0.05}

The special case

e^x

has derivative

e^x

. More precisely, it is the unique solution of

y'=y

with $y(0)=1$ ; without that initial condition, every

Ce^x

solves the same differential equation. Constant proportional change, rather than constant dollar change, is the useful pattern to recognise.

For a fuller treatment of exponent laws, logarithmic inverses and rate calculations, see Exponentials and Logarithms (public preview; full lesson requires membership).

Sources

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals, 6th ed., §§1.5–1.6 and §3.1: exponential functions, logarithms and derivatives.
Hull, J. C. Options, Futures, and Other Derivatives, 9th Global Edition, §4.2, equation (4.2): continuously compounded accumulation.
OpenStax, Calculus Volume 1, §1.5: Exponential and Logarithmic Functions and §3.9: Derivatives of Exponential and Logarithmic Functions.

Equal steps, equal multipliers

A cash balance with an exponential clock

Why the natural base appears in calculus

Sources

Related terms

Exponent

Exponential decay

Exponential growth

Logarithm