Textbook Question
Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?
Ch. 8 - Sequences, Induction, and Probability
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 9, Problem 87
Solve: log2 (x+9) — log2 x = 1.
Verified step by step guidance1
Recall the logarithmic property that states: \(\log_a b - \log_a c = \log_a \left( \frac{b}{c} \right)\), which allows us to combine the difference of two logs with the same base into a single log of a quotient.
Apply this property to the equation \(\log_2 (x+9) - \log_2 x = 1\) to rewrite it as \(\log_2 \left( \frac{x+9}{x} \right) = 1\).
Rewrite the logarithmic equation in its equivalent exponential form: \(\frac{x+9}{x} = 2^1\).
Simplify the right side to get \(\frac{x+9}{x} = 2\), then multiply both sides by \(x\) to clear the denominator, resulting in \(x + 9 = 2x\).
Solve the linear equation \(x + 9 = 2x\) by isolating \(x\), and then check the solution to ensure it does not make any logarithm argument negative or zero.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Logarithmic properties, such as the subtraction rule log_b(A) - log_b(B) = log_b(A/B), allow simplification of expressions involving logs. This property is essential for combining or breaking down logarithmic terms to solve equations.
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Change of Base Property
Definition of Logarithms
A logarithm log_b(A) answers the question: to what power must the base b be raised to get A? Understanding this helps convert logarithmic equations into exponential form, which is often easier to solve.
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Domain Restrictions in Logarithmic Functions
The argument of a logarithm must be positive, so expressions inside logs (like x and x+9) must be greater than zero. Recognizing domain restrictions ensures solutions are valid within the function's domain.
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Related Practice
Textbook Question
Solve: .
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Textbook Question
In Exercises 81–85, use a calculator's factorial key to evaluate each expression.
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Textbook Question
Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?
Textbook Question
Write an equation in point-slope form and slope-intercept form for the line passing through (-2, -6) and perpendicular to the line whose equation is x − 3y+ 9 = 0.
Textbook Question
Graph: f(x) = -2(x − 1)² (x + 3).