Lesson Objectives

- Demonstrate an understanding of the rules of exponents
- Learn how to graph an exponential function
- Learn how to solve an exponential equation

## How to Graph an Exponential Function

An exponential function is of the form:

f(x) = a

a > 0

a ≠ 1

x is any real number

When we think about the graph of f(x) = a

Example 1: Sketch the graph of each. $$f(x) = 3^x$$ Let's create a table with some ordered pairs:

Now we can plot the points on the coordinate plane and connect the points using a smooth curve. As the graph moves from right to left, it approaches the x-axis but does not touch it.

f(x) = a

^{x}a > 0

a ≠ 1

x is any real number

When we think about the graph of f(x) = a

^{x}:- (0,1) is on the graph
- Any non-zero number raised to the power of 0 is 1

- The graph approaches the x-axis, but will never touch it
- It has a horizontal asymptote at y = 0

- The domain consists of all real numbers or the interval: (-∞, ∞)
- The range consists of all positive real numbers, or the interval: (0, ∞)
- When a > 1, the graph rises from left to right
- When 0 < a < 1, the graph falls from left to right

Example 1: Sketch the graph of each. $$f(x) = 3^x$$ Let's create a table with some ordered pairs:

x | y | (x, y) |
---|---|---|

-2 | 1/9 | (-2, 1/9) |

-1 | 1/3 | (-1, 1/3) |

0 | 1 | (0, 1) |

1 | 3 | (1, 3) |

2 | 9 | (2, 9) |

## How to Solve an Exponential Equation

An exponential equation is an equation with a variable in the exponent.

f(x) = a

for a > 0, a ≠ 1

To solve an exponential equation with like bases on each side, we use the following rule:

if: a

then: x = y

In simpler cases, we can use the above rule to solve exponential equations. In other cases, we will need to rely on logarithms, which we will learn about over the course of the next few lessons. To solve an exponential equation with like bases, we can use the following steps.

Example 2: Solve each equation. $$3^{2x + 1}=27$$ Step 1) Make sure each side has the same base.

In this case, the left side has a base of 3, while the right side has a base of 27. Using the rules of exponents, we can rewrite 27 as 3

In this case, the exponents are simplified.

Step 3) Set the exponents equal. $$2x + 1=3$$ Step 4) Solve the resulting equation. $$2x + 1=3$$ $$2x=2$$ $$x=1$$ Step 5) Check $$3^{2x + 1}=3^3$$ $$3^{2(1) + 1}=3^3$$ $$3^{3}=3^3$$ $$\require{color}27=27 \hspace{.2em}\color{green}{✔}$$ Example 3: Solve each equation. $$25^{-3x-3}\cdot 5^{-2x - 2}=625$$ Step 1) Make sure each side has the same base.

In this case, we will write each base as 5, using the rules of exponents. $$5^{2(-3x-3)}\cdot 5^{-2x - 2}=5^4$$ $$5^{-6x - 6}\cdot 5^{-2x - 2}=5^4$$ Step 2) Simplify the exponents.

On the left side of the equation, we can use our product rule for exponents. $$5^{-6x - 6 - 2x - 2}=5^4$$ $$5^{-8x - 8}=5^4$$ Step 3) Set the exponents equal. $$-8x - 8=4$$ Step 4) Solve the resulting equation. $$-8x - 8=4$$ $$-8x=12$$ $$x=-\frac{12}{8}=-\frac{3}{2}$$ Step 5) Check $$25^{-3\cdot -\frac{3}{2}- 3}\cdot 5^{-2 \cdot -\frac{3}{2}- 2}=625$$ $$25^{\frac{9}{2}- 3}\cdot 5^{3 - 2}=625$$ $$25^{\frac{9}{2}- \frac{6}{2}}\cdot 5^1=625$$ $$25^{\frac{3}{2}}\cdot 5^{1}=625$$ $$125 \cdot 5=625$$ $$625=625 \hspace{.2em}\color{green}{✔}$$

f(x) = a

^{x}for a > 0, a ≠ 1

To solve an exponential equation with like bases on each side, we use the following rule:

if: a

^{x}= a^{y}then: x = y

In simpler cases, we can use the above rule to solve exponential equations. In other cases, we will need to rely on logarithms, which we will learn about over the course of the next few lessons. To solve an exponential equation with like bases, we can use the following steps.

- Make sure each side has the same base
- In some cases, we can rewrite our expression using the rules of exponents

- Simplify the exponents
- Set the exponents equal
- Solve the resulting equation
- Check

Example 2: Solve each equation. $$3^{2x + 1}=27$$ Step 1) Make sure each side has the same base.

In this case, the left side has a base of 3, while the right side has a base of 27. Using the rules of exponents, we can rewrite 27 as 3

^{3}. $$3^{2x + 1}=3^3$$ Step 2) Simplify the exponents.In this case, the exponents are simplified.

Step 3) Set the exponents equal. $$2x + 1=3$$ Step 4) Solve the resulting equation. $$2x + 1=3$$ $$2x=2$$ $$x=1$$ Step 5) Check $$3^{2x + 1}=3^3$$ $$3^{2(1) + 1}=3^3$$ $$3^{3}=3^3$$ $$\require{color}27=27 \hspace{.2em}\color{green}{✔}$$ Example 3: Solve each equation. $$25^{-3x-3}\cdot 5^{-2x - 2}=625$$ Step 1) Make sure each side has the same base.

In this case, we will write each base as 5, using the rules of exponents. $$5^{2(-3x-3)}\cdot 5^{-2x - 2}=5^4$$ $$5^{-6x - 6}\cdot 5^{-2x - 2}=5^4$$ Step 2) Simplify the exponents.

On the left side of the equation, we can use our product rule for exponents. $$5^{-6x - 6 - 2x - 2}=5^4$$ $$5^{-8x - 8}=5^4$$ Step 3) Set the exponents equal. $$-8x - 8=4$$ Step 4) Solve the resulting equation. $$-8x - 8=4$$ $$-8x=12$$ $$x=-\frac{12}{8}=-\frac{3}{2}$$ Step 5) Check $$25^{-3\cdot -\frac{3}{2}- 3}\cdot 5^{-2 \cdot -\frac{3}{2}- 2}=625$$ $$25^{\frac{9}{2}- 3}\cdot 5^{3 - 2}=625$$ $$25^{\frac{9}{2}- \frac{6}{2}}\cdot 5^1=625$$ $$25^{\frac{3}{2}}\cdot 5^{1}=625$$ $$125 \cdot 5=625$$ $$625=625 \hspace{.2em}\color{green}{✔}$$

#### Skills Check:

Example #1

Solve each equation. $$4 \cdot \left(\frac{1}{8}\right)^{x}=2^{3}$$

Please choose the best answer.

A

$$x=-1$$

B

$$x=-2$$

C

$$x=-\frac{1}{3}$$

D

$$x=-\frac{5}{7}$$

E

$$x=10$$

Example #2

Solve each equation. $$64^{-x + 2}\cdot \left(\frac{1}{2}\right)^{-2x}=8$$

Please choose the best answer.

A

$$x=9$$

B

$$x=\frac{9}{4}$$

C

$$x=8$$

D

$$x=-4$$

E

$$x=\frac{5}{6}$$

Example #3

Solve each equation. $$81 \cdot 243^{-3x}=27^{-3x}$$

Please choose the best answer.

A

$$x=-\frac{9}{2}$$

B

$$x=7$$

C

$$x=-6$$

D

$$x=-1$$

E

$$x=\frac{2}{3}$$

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