Exponential Functions: Definition, Graphs, Growth and Decay

An exponential function has the form ( f(x) = acdot b^x ) with ( b > 0 ) and ( b neq 1 ): the variable is in the exponent. Its graph passes through ( (0, a) ), has the horizontal asymptote ( y = 0 ), rises when ( b > 1 ) (growth) and falls when ( 0 < b < 1 ) (decay). The most important base is ( e approx 2.71828 ), giving the growth model ( P = P_0e^{kt} ).

An exponential function has the variable in the exponent:

$$f(x) = acdot b^x, qquad b > 0, b neq 1$$

Here a is the starting value (the y-intercept) and b is the base, the factor the output is multiplied by each time x increases by 1. Compare ( x^2 ), where the variable is the base, with ( 2^x ), where it’s the exponent. The difference is enormous: at ( x = 20 ), ( x^2 = 400 ) but ( 2^x = 1{,}048{,}576 ).

Exponent rules (exponente)

Exponential functions follow the exponent laws (explained in full in exponent rules):

Rule	Formula	Example
Product	( b^m b^n = b^{m+n} )	( 2^3 cdot 2^4 = 2^7 )
Quotient	( frac{b^m}{b^n} = b^{m-n} )	( frac{5^6}{5^2} = 5^4 )
Power	( (b^m)^n = b^{mn} )	( (3^2)^3 = 3^6 )
Zero	( b^0 = 1 )	( 7^0 = 1 )
Negative	( b^{-n} = frac{1}{b^n} )	( 2^{-3} = frac18 )
Fractional	( b^{1/n} = sqrt[n]{b} )	( 8^{1/3} = 2 )

The graph of an exponential function

Growth: b > 1

( y = 2^x ):

x	−2	−1	0	1	2	3
( 2^x )	( frac14 )	( frac12 )	1	2	4	8

• Passes through ( (0, 1) ) and ( (1, 2) ).
• Always positive: the range is ( (0, infty) ) and the domain is all real numbers.
• Horizontal asymptote ( y = 0 ) on the left; the graph gets close to the x-axis but never touches it.
• Increases, and gets steeper and steeper.

Decay: 0 < b < 1

( y = left(frac12right)^x = 2^{-x} ) is the mirror image of ( 2^x ) across the y-axis. It still passes through ( (0, 1) ) and has asymptote ( y = 0 ), but now on the right, and it decreases.

Transformations

( y = acdot b^{x – h} + k ) shifts the graph right h and up k, and stretches it by a. The asymptote moves to ( y = k ).

Example 1. ( y = 3cdot 2^x – 4 ) has y-intercept ( 3 – 4 = -1 ), asymptote ( y = -4 ), and range ( (-4, infty) ).

Plot an exponential and see the asymptote (the graph below works as an exponential equation grapher). Try <code>(1/2)^x</code> or <code>3*2^x – 4</code>:

The number e

The most important base is ( e approx 2.718281828 ). It arises from continuous growth:

$$e = lim_{ntoinfty}left(1 + frac1nright)^n$$

If you earn 100% interest per year, compounded n times a year, $1 grows to ( left(1 + frac1nright)^n ) dollars. Compounding yearly gives $2, monthly about $2.613, daily about $2.7146, and continuously exactly ( e ) dollars.

( f(x) = e^x ) is called the exponential function. Its special property is that it is its own derivative: ( frac{d}{dx}e^x = e^x ). The slope of its graph at every point equals its height. See the derivative of e^(2x) for how the chain rule extends this.

Exponential growth and decay models

$$P(t) = P_0e^{kt}$$

Here ( P_0 ) is the initial amount and k is the continuous growth rate: positive for growth, negative for decay. Equivalently, ( P(t) = P_0 b^t ) with ( b = e^k ).

Example 2: population growth. A town of 12,000 grows by 3% per year: ( P(t) = 12000(1.03)^t ). After 10 years, ( P(10) = 12000(1.03)^{10} approx 16{,}127 ).

Example 3: half-life. A drug’s half-life is 5 hours. From an 80 mg dose, the amount left after t hours is ( 80left(frac12right)^{t/5} ). After 12 hours: ( 80(0.5)^{2.4} approx 15.16 ) mg.

Example 4: doubling time. At a continuous rate k, doubling time is ( frac{ln 2}{k} ). For 5%, that’s about 13.86 years.

Try a model in the calculator: starting amount, rate k and time t.

The full-page exponential growth calculator also does decay and half-life.

The half-life equation

A quantity that halves every ( t_{1/2} ) time units follows

$$N(t) = N_0left(frac12right)^{t/t_{1/2}} = N_0e^{-kt}, qquad k = frac{ln 2}{t_{1/2}}$$

Example. Carbon-14 has a half-life of about 5730 years. A sample with 25% of its original carbon-14 has gone through ( log_{1/2} 0.25 = 2 ) half-lives, so it’s about 11,460 years old. To find the half-life from data, solve ( frac12 = e^{-kt_{1/2}} ): ( t_{1/2} = frac{ln 2}{k} ). The exponential growth calculator reports half-life directly when you enter a negative k.

Compound interest

$$A = Pleft(1 + frac{r}{n}right)^{nt} qquadtext{and}qquad A = Pe^{rt} text{ (continuous)}$$

Example 5. $1000 at 5% for 10 years:

Compounding	Amount
yearly, ( 1000(1.05)^{10} )	$1628.89
continuously, ( 1000e^{0.5} )	$1648.72

Example 6. $5000 at 4% compounded monthly for 3 years: ( 5000left(1 + frac{0.04}{12}right)^{36} approx $5636.36 ).

Solving exponential equations

Same base. Rewrite both sides with a common base and set the exponents equal. ( 4^{x+1} = 8^x ) becomes ( 2^{2x+2} = 2^{3x} ), so ( 2x + 2 = 3x ) and ( x = 2 ).

Different bases. Take a log of both sides. ( 3^x = 20 ) gives ( x = frac{ln 20}{ln 3} approx 2.727 ). See logarithm rules and the logarithm calculator.

Example 7. When does the town in Example 2 reach 20,000? Solve ( 12000(1.03)^t = 20000 ): ( 1.03^t = frac53 ), so ( t = frac{ln(5/3)}{ln 1.03} approx 17.3 ) years.

Exponential vs linear growth

	Linear ( y = mx + b )	Exponential ( y = acdot b^x )
Each step in x	adds m	multiplies by b
Table test	constant differences	constant ratios
Long run	steady	eventually overtakes any linear function

Exponential functions in calculus

• Derivatives. ( frac{d}{dx}e^x = e^x ) and ( frac{d}{dx}b^x = b^xln b ).
• Integrals. ( int e^x,dx = e^x + C ), and ( int xe^x,dx ) needs integration by parts (see the integral of xe^x).
• Differential equations. ( frac{dP}{dt} = kP ) has solution ( P = P_0e^{kt} ). That’s why exponentials model anything whose growth is proportional to its size; see differential equations.
• Limits. Exponentials beat every power: ( lim_{xtoinfty}frac{x^{100}}{e^x} = 0 ), which L’Hôpital’s rule proves.

Common mistakes

• ( 2^x ) vs ( x^2 ). The first is exponential, the second is a polynomial.
• Distributing exponents over sums. ( (a + b)^x neq a^x + b^x ).
• Negative base. Exponential functions need ( b > 0 ).
• Percent vs factor. 3% growth is a factor of 1.03, not 3 or 0.03. 3% decay is 0.97.
• Thinking the graph hits zero. ( b^x > 0 ) for every x.

Practice problems

1. Evaluate ( 16^{3/4} ).
2. Find the y-intercept and asymptote of ( y = 5cdot 3^x + 2 ).
3. Solve ( 9^x = 27 ).
4. A car worth $25,000 loses 15% of its value each year. Find its value after 4 years.
5. Solve ( e^{3x} = 10 ).
6. How much is $2000 after 5 years at 6% compounded continuously?

Answers: (1) 8; (2) y-intercept 7, asymptote ( y = 2 ); (3) ( x = frac32 ); (4) about $13,050.16; (5) ( x = frac{ln 10}{3} approx 0.7675 ); (6) about $2699.72.

FAQ

What is an exponential function?

A function of the form ( f(x) = acdot b^x ) with a positive base ( b neq 1 ), where the variable is in the exponent.

What does an exponential graph look like?

A curve that passes through ( (0, a) ), hugs a horizontal asymptote on one side, and rises (or falls) ever more steeply on the other.

What is the difference between exponential growth and decay?

Growth has ( b > 1 ) (or ( k > 0 )) and increases; decay has ( 0 < b < 1 ) (or ( k < 0 )) and decreases toward the asymptote.

Why is e used so often?

Because ( e^x ) is its own derivative, which makes it the natural base for continuous growth and for calculus.

How do you solve for x in an exponent?

Take the logarithm of both sides, then use the power rule ( log b^x = xlog b ).

Further reading

• Exponential and Logarithm Functions (Paul’s Online Math Notes) — definitions, graphs and the natural exponential function.
• Exponential function (Wikipedia) — properties of ( e^x ), its series and applications.