Exponential and Logarithmic Functions: Learn It 1

  • Work with exponential functions to find their values
  • Recognize logarithmic functions, explore their relationship with exponential functions, and change their bases
  • Identify hyperbolic functions their graphs, and understand their fundamental identities

Exponential Functions

Exponential functions arise in many applications. One common example is population growth.

If a population starts with [latex]P_0[/latex] individuals and then grows at an annual rate of [latex]2\%[/latex], its population after [latex]1[/latex] year is

[latex]P(1)=P_0+0.02P_0=P_0(1+0.02)=P_0(1.02)[/latex]

Its population after [latex]2[/latex] years is

[latex]P(2)=P(1)+0.02P(1)=P(1)(1.02)=P_0(1.02)^2[/latex]

In general, its population after [latex]t[/latex] years is

[latex]P(t)=P_0(1.02)^t[/latex],

which is an exponential function.

More generally, any function of the form [latex]f(x)=b^x[/latex], where [latex]b>0, \, b \ne 1[/latex], is an exponential function with base [latex]b[/latex] and exponent [latex]x[/latex]. Exponential functions have constant bases and variable exponents.

exponential function

For any real number [latex]x[/latex], an exponential function is a function with the form

[latex]f(x)=ab^x[/latex]

where,

  • [latex]a[/latex] is a non-zero real number called the initial value and
  • [latex]b[/latex] is any positive real number ([latex]b>0[/latex]) such that [latex]b≠1[/latex].

Why do we limit the base [latex]b[/latex] to positive values?

To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:

  • Let [latex]b=−9[/latex] and [latex]x=\frac{1}{2}[/latex]. Then [latex]f(x)=f(\frac{1}{2})=(−9)^\frac{1}{2}=\sqrt{−9}[/latex], which is not a real number.

Why do we limit the base to positive values other than [latex]1[/latex]?

Because base [latex]1[/latex] results in the constant function. Observe what happens if the base is [latex]1[/latex]:

  • Let [latex]b=1[/latex]. Then [latex]f(x)=1^x=1[/latex] for any value of [latex]x[/latex].

Note that a function of the form [latex]f(x)=x^b[/latex] for some constant [latex]b[/latex] is not an exponential function but a power function.

To see the difference between an exponential function and a power function, we can compare the functions [latex]y=x^2[/latex] and [latex]y=2^x[/latex].

In the table below, we see that both [latex]2^x[/latex] and [latex]x^2[/latex] approach infinity as [latex]x \to \infty[/latex]. Eventually, however, [latex]2^x[/latex] becomes larger than [latex]x^2[/latex] and grows more rapidly as [latex]x \to \infty[/latex]. In the opposite direction, as [latex]x \to −\infty, \, x^2 \to \infty[/latex], whereas [latex]2^x \to 0[/latex]. The line [latex]y=0[/latex] is a horizontal asymptote for [latex]y=2^x[/latex].

Values of [latex]x^2[/latex] and [latex]2^x[/latex]
[latex]\mathbf{x}[/latex] [latex]\mathbf{x^2}[/latex] [latex]\mathbf{2^x}[/latex]
[latex]-3[/latex] [latex]9[/latex] [latex]1/8[/latex]
[latex]-2[/latex] [latex]4[/latex] [latex]1/4[/latex]
[latex]-1[/latex] [latex]1[/latex] [latex]1/2[/latex]
[latex]0[/latex] [latex]0[/latex] [latex]1[/latex]
[latex]1[/latex] [latex]1[/latex] [latex]2[/latex]
[latex]2[/latex] [latex]4[/latex] [latex]4[/latex]
[latex]3[/latex] [latex]9[/latex] [latex]8[/latex]
[latex]4[/latex] [latex]16[/latex] [latex]16[/latex]
[latex]5[/latex] [latex]25[/latex] [latex]32[/latex]
[latex]6[/latex] [latex]36[/latex] [latex]64[/latex]
An image of a graph. The x axis runs from -10 to 10 and the y axis runs from 0 to 50. The graph is of two functions. The first function is “y = x squared”, which is a parabola. The function decreases until it hits the origin and then begins increasing. The second function is “y = 2 to the power of x”, which starts slightly above the x axis, and begins increasing very rapidly, more rapidly than the first function.
Figure 1. Both [latex]2^x[/latex] and [latex]x^2[/latex] approach infinity as [latex]x \to \infty[/latex], but [latex]2^x[/latex] grows more rapidly than [latex]x^2[/latex]. As [latex]x \to −\infty, \, x^2 \to \infty[/latex], whereas [latex]2^x \to 0[/latex].
Arrow Notation
Symbol Meaning
[latex]x\to \infty[/latex] [latex]x[/latex] approaches infinity ([latex]x[/latex] increases without bound)
[latex]x\to -\infty[/latex] [latex]x[/latex] approaches negative infinity ([latex]x[/latex] decreases without bound)
[latex]f\left(x\right)\to \infty[/latex] the output approaches infinity (the output increases without bound)
[latex]f\left(x\right)\to -\infty[/latex] the output approaches negative infinity (the output decreases without bound)
[latex]f\left(x\right)\to a[/latex] the output approaches [latex]a[/latex]