"

3.3: Exponential Functions

Learning Objectives

  1. Evaluate exponential functions.
  2. List the properties of exponential functions and their graphs.
  3. Solve exponential equations.

Evaluating Exponential Functions

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

[latex]f\left(x\right)=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 such that [latex]b\neq1[/latex].
  • The domain of [latex]f[/latex] is all real numbers.
  • The range of [latex]f[/latex] is all positive real numbers if [latex]a>0[/latex].
  • The range of [latex]f[/latex] is all negative real numbers if [latex]a<0[/latex].
  • The y-intercept is [latex]\left(0,\;a\right)[/latex], and the horizontal asymptote is [latex]y=0[/latex].
Inversilina Character

It is important to correctly identify the base of an exponential function. In the standard form [latex]f\left(x\right)=ab^x[/latex], the variable [latex]x[/latex] is the exponent, [latex]b[/latex] represents the base, and [latex]a[/latex] is a constant coefficient. Please refer to the attached image for clarification.

[latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol f}\mathbf{\color[rgb]{1.0, 1.0, 1.0}\left(x\right)}{\color[rgb]{1.0, 1.0, 1.0}\mathbf=}{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol a}{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol b}^{\color[rgb]{1.0, 1.0, 1.0}\mathbf x}[/latex]
[latex]a[/latex] Coefficient
[latex]b[/latex] Base
[latex]x[/latex] Power/exponent

Example 3.3-1-1: Identifying Exponential Functions: Identify the base of the exponential function. And find the value for part c and d

  1. [latex]f\left(x\right)=4^{3\left(x-2\right)}[/latex]
  2. [latex]f\left(x\right)=0.35e^{3+x}[/latex]
  3. [latex]f\left(x\right)=-\frac13^2[/latex]
  4. [latex]f\left(x\right)=\left(-\frac13\right)^2[/latex]

Inverselina Character Key

Example 3.3-1-1: Identifying Exponential Functions: Identify the base of the exponential function. And find the value for part c and d

  1. [latex]f\left(x\right)=4^{3\left(x-2\right)}[/latex]
  2. [latex]f\left(x\right)=0.35e^{3+x}[/latex]
  3. [latex]f\left(x\right)=-\frac13^2[/latex]
  4. [latex]f\left(x\right)=\left(-\frac13\right)^2[/latex]
  1. [latex]f\left(x\right)=4^{3\left(x-2\right)}[/latex] the base is [latex]4[/latex].
  2. [latex]f\left(x\right)=0.35e^{3+x}[/latex] the base is [latex]e[/latex].
  3. [latex]f\left(x\right)=-\frac13^2[/latex] the base is [latex]\frac13[/latex].

[latex]f\left(x\right)=-\cdot\frac13\cdot\frac13=-\frac19[/latex]

  1. [latex]f\left(x\right)=\left(-\frac13\right)^2[/latex] the base is [latex]-\frac13[/latex].

[latex]f\left(x\right)=\left(-\frac13\right)\left(-\frac13\right)=\frac19[/latex]

Your Turn

Practice 3.3-1-1

Properties of Exponential Functions

[latex]f\left(x\right)=a^x\;or\;y=a^x,\;where\;a>0,\;a\neq1,\;and\;x\in\mathbb{R}\;\left(Domain\right),\;f\left(x\right)>0\left(range\right).[/latex] The shape of the graph of [latex]f\left(x\right)=a^x[/latex] is given by:

  • If [latex]0<[/latex] [latex]a<1[/latex], then the function decreases from left to right.

A graph of a decreasing exponential function on a Cartesian coordinate system. The x and y axes intersect at the origin. A blue curve starts high in the second quadrant, passes through the point (0, 1), which is marked with a black dot and labeled, and then decreases asymptotically towards the positive x-axis in the first quadrant. The label '0 < a < 1' in blue indicates that the base 'a' of the exponential function is between 0 and 1, which accounts for the decreasing nature of the graph.

  • If [latex]a>1[/latex], then the function increases from left to right.

A graph of an increasing exponential function on a Cartesian coordinate system. The x and y axes intersect at the origin. A blue curve starts close to the negative x-axis in the third quadrant, passes through the point (0, 1), which is marked with a black dot and labeled, and then increases rapidly into the first quadrant. The label 'a > 1' in blue indicates that the base 'a' of the exponential function is greater than 1, which accounts for the increasing nature of the graph.

  • Both always pass [latex](0,1)[/latex].
  • A graph of exponential function [latex]f\left(x\right)=a^x[/latex] or [latex]y=a^x[/latex], where [latex]a>0,\;a\neq1[/latex], has no x-intercept.

Example 3.3-2-1: Graph the function.

[latex]f\left(x\right)=8^x[/latex]

Inverselina Character Key

Example 3.3-2-1: Graph the function.

[latex]f\left(x\right)=8^x[/latex]

Based on the property, we know that x can be any value; thus, we can use a table to graph the function.

[latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol x}[/latex] [latex]{\color[rgb]{1.0, 1.0, 1.0}\boldsymbol f}\mathbf{\color[rgb]{1.0, 1.0, 1.0}\left(x\right)}[/latex] [latex]\mathbf{\color[rgb]{1.0, 1.0, 1.0}\left({x,\;y}\right)}[/latex]
-1 [latex]8^{-1}=\frac18=0.125[/latex] [latex]\left(-1,\;0.125\right)[/latex]
0 [latex]8^0=1[/latex] [latex]\left(0,\;1\right)[/latex]
1 [latex]8^1=8[/latex] [latex]\left(1,\;8\right)[/latex]

A graph of an exponential function on a Cartesian coordinate system. The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10, with grid lines at integer values. A red curve represents an increasing exponential function. It passes through three blue points, which are labeled with their coordinates: (-1, 0.125), (0, 1), and (1, 8). The curve approaches the negative x-axis asymptotically.

Master L3 Character

Tips for graphing problem works for ALL:

  1. Identify the Domain – Determine which x-values are valid for the function.
  2. Choose x-Values – Select a few values within the domain.
  3. Calculate y-Values – Plug x-values into the function to find corresponding y-values.
  4. Plot Points – Write each (x, y) as coordinates and plot them.
  5. Draw and Extend – Connect the points smoothly and extend the graph based on function properties.

Your Turn

Practice 3.3-2-1

Evaluate composite functions

Using the One-to-One Property to Solve Exponential Equations

[latex]If\;a^u=a^v,\;then\;u=v.[/latex]

  1. Rewrite the equation in the form [latex]a^u=a^v[/latex].
  2. Set [latex]u=v[/latex].
  3. Solve for the variable.

Useful Rules:

[latex]b=b^1[/latex] ([latex]b[/latex] is any number) when power is [latex]1[/latex], we do not write it out.
[latex]b^0=1[/latex] any number with power [latex]0[/latex] is [latex]1[/latex].
[latex]b^m\cdot b^n=b^{m+n}[/latex] multiply powers of the same base, add exponents.
[latex]\frac{b^m}{b^n}=b^{m-n}[/latex] divide powers of the same base, subtract (numerator exponent minus denominator exponent).
[latex]\left(b^m\right)^n=b^{m\cdot n}[/latex] multiple powers, multiple powers.
[latex]b^{-n}=\frac1{b^n}\;and\;\frac1{b^{-n}}=b^n[/latex] negative powers, reciprocal base, change power to positive.
[latex]\left(a\cdot b\right)^n=a^n\cdot b^n[/latex] raise a product to a power, raise power for each base.
[latex]\left(\frac ab\right)^n=\frac{a^n}{b^n}[/latex] raise fraction to a power, raise power for numerator and denominator.
[latex]\left(\frac ab\right)^{-n}=\left(\frac ba\right)^n[/latex] fraction raised to a negative power, reciprocal fraction change power to positive.

Example 3.3-3-1: Solve Elementary Exponential Equations.

[latex]3^{x+1}=81[/latex]

Inverselina Character Key

Example 3.3-3-1: Solve Elementary Exponential Equations.

The exponential equation 3 to the power x+1 = 81 is shown. A teal arrow points from the text 'Prime Number Base' to the base of the exponential term, which is '3'.

Step 1: To have [latex]a^u=a^v[/latex] we need to rewriting [latex]81[/latex] with Base [latex]3[/latex], We choose base [latex]3[/latex] because the expression on the right side involves base [latex]3[/latex]. Additionally, [latex]3[/latex] is a prime number, meaning it cannot be reduced to any other integer base.

A factor tree illustrating the prime factorization of the number 81. Starting at the top, the number 81 has two downward arrows leading to its factors: 3 and 27. The number 3 is enclosed in a pink circle, indicating it is a prime factor. The number 27 has two downward arrows leading to its factors: 3 and 9. The number 3 is again enclosed in a pink circle as a prime factor. The number 9 has two downward arrows leading to its factors: 3 and 3. Both of these 3s are enclosed in pink circles, indicating they are prime factors. Therefore, the prime factorization of 81 is 3 × 3 × 3 × 3, or 3 to the power 4.

[latex]81={\color[rgb]{0.5, 0.0, 0.5}\mathbf3}^{\color[rgb]{0.5, 0.0, 0.5}\mathbf4}[/latex]

Thus:

[latex]3^{x+1}=81[/latex]

[latex]3^{x+1}={\color[rgb]{0.5, 0.0, 0.5}\mathbf3}^{\color[rgb]{0.5, 0.0, 0.5}\mathbf4}[/latex]

Step 2: Since left and right bases are the same, we just need to compare the power.

[latex]x+1=4[/latex]

Step 3: Solve for the variable.

[latex]x=4-1[/latex]

[latex]x=3[/latex]

Example 3.3-3-2: Solve Elementary Exponential Equations.

[latex]7^{x+3}=\frac17[/latex]

Inverselina Character Key

Example 3.3-3-2: Solve Elementary Exponential Equations.

The exponential equation 7^(x+3) = 1/7 is shown. A teal arrow points from the text 'Prime Number Base' to the base of the exponential term, which is '7'.

Step 1: To have [latex]a^u=a^v[/latex] we need to flip [latex]\frac17[/latex]

Based on the rule

[latex]\left(\frac ab\right)^{-n}=\left(\frac ba\right)^n[/latex]

[latex]\frac17=\left(\frac17\right)^1=\left(\frac17\right)^{-1}=7^{-1}[/latex]

Thus:

[latex]\frac17=7^{-1}[/latex]

[latex]7^{x+3}={\color[rgb]{0.5, 0.0, 0.5}\frac{\mathbf1}{\mathbf7}}[/latex]

[latex]7^{x+3}={\color[rgb]{0.5, 0.0, 0.5}\mathbf7}^{\color[rgb]{0.5, 0.0, 0.5}{\boldsymbol-\mathbf1}}[/latex]

Step 2: Since left and right bases are the same, we just need to compare the power.

[latex]x+3=-1[/latex]

Step 3: Solve for the variable.

[latex]x=-1-3[/latex]

[latex]x=-4[/latex]

Example 3.3-3-3: Solve Elementary Exponential Equations.

[latex]8^{x+2}=4^{x-3}[/latex]

Inverselina Character Key

Example 3.3-3-3: Solve Elementary Exponential Equations.

The exponential equation 8 to the power (x+2) = 4 to the power (x-3) is shown. A teal arrow points to the base of the left exponential term, '8', with the label 'Non-Prime Number Base' below it. A magenta arrow points to the base of the right exponential term, '4', also with the label 'Non-Prime Number Base' below it.

Step 1: To have [latex]a^u=a^v[/latex] we can first reduce both bases to prime numbers.

Two factor trees are shown side-by-side. The factor tree on the left, in teal, starts with the number 8. Two arrows branch down to its factors, 2 and 4. The 2 is at the end of its branch. From the 4, two more arrows branch down to its factors, 2 and 2, which are at the end of their branches. This shows the prime factorization of 8 as 2 × 2 × 2. The factor tree on the right, in magenta, starts with the number 4. Two arrows branch down to its factors, 2 and 2, which are at the end of their branches. This shows the prime factorization of 4 as 2 × 2.

Thus: [latex]8=2^3,\;4=2^2[/latex]

[latex]\left({\color[rgb]{0.0, 0.44, 0.73}\mathbf2}^{\color[rgb]{0.0, 0.44, 0.73}\mathbf3}\right)^{x+2}=\left({\color[rgb]{0.5, 0.0, 0.5}\mathbf2}^{\color[rgb]{0.5, 0.0, 0.5}\mathbf2}\right)^{x-3}[/latex]

Simplify the equation by using rule

[latex]\left(b^m\right)^n=b^{m\cdot n}[/latex]

[latex]{\color[rgb]{0.0, 0.44, 0.73}\mathbf2}^{\left({\color[rgb]{0.0, 0.44, 0.73}\mathbf3}\right)\left(x+2\right)}={\color[rgb]{0.5, 0.0, 0.5}\mathbf2}^{\left({\color[rgb]{0.5, 0.0, 0.5}\mathbf2}\right)\left(x-3\right)}[/latex]

[latex]{\color[rgb]{0.0, 0.44, 0.73}\mathbf2}^{{\color[rgb]{0.0, 0.44, 0.73}\mathbf3}x+6}={\color[rgb]{0.5, 0.0, 0.5}\mathbf2}^{{\color[rgb]{0.5, 0.0, 0.5}\mathbf2}x-6}[/latex]

Step 2: Since left and right bases are the same, we just need to compare the power.

[latex]3x+6=2x-6[/latex]

Step 3: Solve for the variable.

[latex]3x-2x+6=-6[/latex]

[latex]x+6=-6[/latex]

[latex]x=-6-6[/latex]

[latex]x=-12[/latex]

Your Turn

Practice 3.3-3-1

Licenses and Attribution
CC Licensed Content

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Math Mastery Manual Copyright © by Linglin (Niki) Liu is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book