Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Solving Exponential and Logarithmic Equations

College Algebra

6. Exponential & Logarithmic Functions

Solving Exponential and Logarithmic Equations

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2:24 minutes

Problem 94

Textbook Question

In Exercises 93–102, solve each equation. 3^x+2 ⋅ 3^x=81

Verified step by step guidance

Recognize that the equation can be rewritten using the property of exponents: \(3^x + 2 \cdot 3^x = 81\).

Factor out \(3^x\) from the left side of the equation: \(3^x(1 + 2) = 81\).

Simplify the expression inside the parentheses: \(3^x \cdot 3 = 81\).

Divide both sides of the equation by 3 to isolate \(3^x\): \(3^x = \frac{81}{3}\).

Recognize that \(81\) is a power of 3, specifically \(3^4\), and set \(3^x = 3^4\) to solve for \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Equations

Exponential equations involve variables in the exponent, such as 3^x. To solve these equations, one often needs to manipulate the equation to isolate the variable. This can include using properties of exponents, such as the product rule, which states that a^m ⋅ a^n = a^(m+n). Understanding how to work with exponents is crucial for finding the value of x.

Properties of Exponents

The properties of exponents are rules that govern how to simplify expressions involving exponents. Key properties include the product of powers, power of a power, and power of a product. For example, in the equation 3^x + 2 ⋅ 3^x, recognizing that both terms can be combined using the product rule is essential for simplifying the equation effectively.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to solve for variables in the exponent. When an exponential equation is simplified, it may be necessary to apply logarithms to isolate the variable. For instance, if you reach a point where you have an equation like 3^x = a, you can use logarithms to solve for x by rewriting it as x = log_3(a).