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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1:51 minutes

Problem 15

Textbook Question

Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 6^(x−3)/4=√6

Verified step by step guidance

Rewrite the square root of 6 as an exponent: \( \sqrt{6} = 6^{1/2} \).

Express both sides of the equation with the same base: \( \frac{6^{x-3}}{4} = 6^{1/2} \).

Multiply both sides by 4 to isolate the exponential expression: \( 6^{x-3} = 4 \times 6^{1/2} \).

Express 4 as a power of 6 if possible, or leave it as is, and equate the exponents: \( x-3 = 1/2 + \log_6(4) \) if 4 can be expressed as a power of 6.

Solve for \( x \) by isolating it on one side of the equation.

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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 are equations in which variables appear as exponents. To solve these equations, one common method is to express both sides of the equation with the same base. This allows us to equate the exponents directly, simplifying the process of finding the variable's value.

Properties of Exponents

The properties of exponents, such as the product of powers, power of a power, and quotient of powers, are essential for manipulating exponential expressions. For instance, when dividing like bases, we subtract the exponents. Understanding these properties helps in rewriting expressions to facilitate solving equations.

Square Roots and Exponents

The square root of a number can be expressed as an exponent of 1/2. For example, √6 can be rewritten as 6^(1/2). This concept is crucial when solving exponential equations, as it allows us to express both sides of the equation in terms of the same base, enabling the comparison of exponents.