Exponential functions

What it is

An exponential function has the form f(x) = C · aˣ, where C ≠ 0 and the base a is positive and not 1. C is the starting value, f(0) = C, and a is the growth factor: when x increases by 1, the value is multiplied by a.

• f(x) = 2ˣ. f(0) = 1, f(1) = 2, f(2) = 4, f(3) = 8 and f(−1) = 1/2. Each step to the right doubles the value.
• f(x) = 500 · 1.05ˣ. An amount of 500 growing by 5% a year. After 10 years it is 500 · 1.05¹⁰ ≈ 814.45.
• f(x) = 80 · 0.5ˣ. The value halves at each step: 80, 40, 20, 10, 5 …

The graph passes through the point (0, C). For C > 0 the function increases when a > 1 and decreases when 0 < a < 1, and the graph stays above the x-axis, which is a horizontal asymptote. For C < 0 the graph is reflected in the x-axis. The domain is all real numbers.

The conditions on the base have their reasons. With a = 1, f(x) = C is a constant. With a negative base, aˣ has no value for many x: (−2)^(1/2) is not a real number.

Not exponential functions:

• x². This is a power function: x is the base and the exponent is fixed. For x > 0, 2ˣ and x² are equal at x = 2 and x = 4, and after that 2ˣ is larger: 2⁵ = 32 against 5² = 25, and 2¹⁰ = 1,024 against 10² = 100.
• 3x + 1. This is a linear function, which adds 3 at each step instead of multiplying.

Exponential or linear. Both have a starting value and are fixed by two pieces of information, such as two points. The difference is what stays the same at each step. A linear function adds the same number, so the difference f(x + 1) − f(x) is constant. An exponential function multiplies by the same number, so the ratio f(x + 1)/f(x) is constant. The graph is a straight line and a curve respectively.

Both start at 100 and are at 120 after one step. Then the exponential one pulls away.

Why it matters

Exponential functions describe quantities that change by the same proportion at each step: compound interest, radioactive decay, and the growth of a population for as long as nothing slows it down.

• Exponential growth solves y = C · aˣ for the value, the starting value, the growth factor or the number of steps. An increase of p% per step gives a = 1 + p/100, as on the change factor page.
• The exponential equation aˣ = b is solved with logarithms. 2ˣ = 50 gives x = ln 50 / ln 2 ≈ 5.64.
• The half-life is the time it takes for the value to go from C to C/2 when a < 1.
• The derivative of aˣ is aˣ · ln a. It equals the function itself only when ln a = 1, that is, when a is the number e, which is why eˣ has a rule of its own.

Plot 2ˣ, 1.2ˣ and 0.5ˣ together in the graphing calculator, and work out how long a given growth rate takes to double something with the doubling time calculator.

Common misconceptions

• “Exponential growth is fast.” That depends on the base. 100 · 1.01ˣ grows by 1% per step and is 110.46 after 10 steps, while 100 + 20x is 300. In the long run an exponential function with C > 0 and a > 1 still overtakes every linear function.
• “2ˣ and x² are the same kind of function.” In x² the exponent is fixed and x is the base. In 2ˣ the base is fixed and x is the exponent. For x > 4, 2ˣ is larger than x².
• “20% a year for five years makes 100%.” The factor is 1.2⁵ ≈ 2.49, an increase of about 149%.
• “The base can be any number.” It has to be positive and not 1. With a negative base the sign alternates for whole-number x, and for many other x there is no value.
• “A decreasing exponential function reaches 0 in the end.” 80 · 0.5ˣ is 80/1,024 ≈ 0.078 after 10 steps and smaller after more, but it is 0 for no x. The graph approaches the x-axis without reaching it.

Common questions