Textbook Question
n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 73(2.6)^x
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 93
Solve each equation. 52x ⋅ 54x=125
Verified step by step guidance1
Recognize that the equation involves exponential expressions with the same base. The equation is \(5^{2x} \cdot 5^{4x} = 125\).
Use the property of exponents that states when multiplying like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). So rewrite the left side as \(5^{2x + 4x} = 5^{6x}\).
Rewrite the right side, 125, as a power of 5. Since \(125 = 5^3\), the equation becomes \(5^{6x} = 5^3\).
Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(6x = 3\).
Solve the equation \(6x = 3\) for \(x\) by dividing both sides by 6: \(x = \frac{3}{6}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Exponents
This concept involves rules for manipulating expressions with exponents, such as multiplying powers with the same base by adding their exponents. For example, 5^(2x) * 5^(4x) equals 5^(2x + 4x) = 5^(6x). Understanding these properties simplifies solving exponential equations.
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Rational Exponents
Expressing Numbers with the Same Base
To solve exponential equations, it helps to rewrite all terms with the same base. Since 125 can be written as 5^3, the equation 5^(6x) = 125 becomes 5^(6x) = 5^3, allowing us to set the exponents equal to each other for solving.
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Solving Linear Equations
After equating the exponents, the problem reduces to solving a linear equation like 6x = 3. This involves isolating the variable by performing inverse operations, such as dividing both sides by 6, to find the value of x.
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Related Practice
Textbook Question
In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(x + 1) = ln x + ln 1
Textbook Question
In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement.
x log 10x = x2
Textbook Question
Solve each equation. 3x+2 ⋅ 3x=81
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Textbook Question
Evaluate or simplify each expression without using a calculator. eln 125
Textbook Question
n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 6.5(0.43)^x