Learn on PengienVision, Algebra 2Chapter 6: Exponential and Logarithmic Functions

Lesson 6: Exponential and Logarithmic Equations

In this Grade 11 enVision Algebra 2 lesson, students learn to solve exponential and logarithmic equations using the Property of Equality for Exponential Equations, common bases, and the Power Property of Logarithms. The lesson covers techniques such as rewriting equations with a common base, converting exponential equations to logarithmic form, and applying logarithms to solve equations like 3^(x+1) = 5^x where no common base exists. Students also apply these skills to real-world exponential models, including problems involving fire growth and sales data.

Section 1

One-to-One Property

Property

One-to-One Property of Exponential Equations
For $a > 0$ and $a \neq 1$ ,

\operatorname{If}\ a^x = a^y,\ \operatorname{then}\ x = y.

To solve an exponential equation:

Write both sides of the equation with the same base, if possible.
Write a new equation by setting the exponents equal.
Solve the new equation.
Check the solution.

Examples

To solve $4^{x+2} = 64$ , first rewrite $64$ as $4^3$ . The equation becomes $4^{x+2} = 4^3$ . Now, set the exponents equal: $x+2 = 3$ , which gives $x=1$ .
Solve $e^{x^2-3} = e^{2x}$ . Since the bases are both $e$ , we set the exponents equal: $x^2-3 = 2x$ . This gives a quadratic equation $x^2-2x-3=0$ , which factors to $(x-3)(x+1)=0$ . So, $x=3$ or $x=-1$ .
Solve $9^{x} = 27$ . Write both sides with base 3: $(3^2)^x = 3^3$ , which simplifies to $3^{2x} = 3^3$ . Therefore, $2x=3$ , and $x=\frac{3}{2}$ .

Explanation

This property is a powerful shortcut for solving exponential equations. If you can make the bases on both sides of the equation the same, you can ignore the bases and simply set the exponents equal to each other to solve.

Section 2

Solving Exponential Equations

Property

An exponential equation is one in which the variable is part of an exponent. Many exponential equations can be solved by writing both sides of the equation as powers with the same base. In general, if two equivalent powers have the same base, then their exponents must be equal. If $b^m = b^n$ , then $m=n$ (for $b > 0, b \neq 1$ ).

Examples

To solve the equation $3^x = 81$ , rewrite it as $3^x = 3^4$ . Equating the exponents gives the solution $x=4$ .

For the equation $5^{x-1} = 125$ , write it as $5^{x-1} = 5^3$ . Then, solve the simpler equation $x-1 = 3$ to get $x=4$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

One-to-One Property

Property

One-to-One Property of Exponential Equations
For $a > 0$ and $a \neq 1$ ,

\operatorname{If}\ a^x = a^y,\ \operatorname{then}\ x = y.

To solve an exponential equation:

Write both sides of the equation with the same base, if possible.
Write a new equation by setting the exponents equal.
Solve the new equation.
Check the solution.

Examples

To solve $4^{x+2} = 64$ , first rewrite $64$ as $4^3$ . The equation becomes $4^{x+2} = 4^3$ . Now, set the exponents equal: $x+2 = 3$ , which gives $x=1$ .
Solve $e^{x^2-3} = e^{2x}$ . Since the bases are both $e$ , we set the exponents equal: $x^2-3 = 2x$ . This gives a quadratic equation $x^2-2x-3=0$ , which factors to $(x-3)(x+1)=0$ . So, $x=3$ or $x=-1$ .
Solve $9^{x} = 27$ . Write both sides with base 3: $(3^2)^x = 3^3$ , which simplifies to $3^{2x} = 3^3$ . Therefore, $2x=3$ , and $x=\frac{3}{2}$ .

Explanation

Section 2

Solving Exponential Equations

Property

Examples

To solve the equation $3^x = 81$ , rewrite it as $3^x = 3^4$ . Equating the exponents gives the solution $x=4$ .

For the equation $5^{x-1} = 125$ , write it as $5^{x-1} = 5^3$ . Then, solve the simpler equation $x-1 = 3$ to get $x=4$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

One-to-One Property

Property

One-to-One Property of Exponential Equations
For $a > 0$ and $a \neq 1$ ,

\operatorname{If}\ a^x = a^y,\ \operatorname{then}\ x = y.

To solve an exponential equation:

Write both sides of the equation with the same base, if possible.
Write a new equation by setting the exponents equal.
Solve the new equation.
Check the solution.

Examples

To solve $4^{x+2} = 64$ , first rewrite $64$ as $4^3$ . The equation becomes $4^{x+2} = 4^3$ . Now, set the exponents equal: $x+2 = 3$ , which gives $x=1$ .
Solve $e^{x^2-3} = e^{2x}$ . Since the bases are both $e$ , we set the exponents equal: $x^2-3 = 2x$ . This gives a quadratic equation $x^2-2x-3=0$ , which factors to $(x-3)(x+1)=0$ . So, $x=3$ or $x=-1$ .
Solve $9^{x} = 27$ . Write both sides with base 3: $(3^2)^x = 3^3$ , which simplifies to $3^{2x} = 3^3$ . Therefore, $2x=3$ , and $x=\frac{3}{2}$ .

Explanation

Section 2

Solving Exponential Equations

Property

Examples

To solve the equation $3^x = 81$ , rewrite it as $3^x = 3^4$ . Equating the exponents gives the solution $x=4$ .

For the equation $5^{x-1} = 125$ , write it as $5^{x-1} = 5^3$ . Then, solve the simpler equation $x-1 = 3$ to get $x=4$ .