Textbook Question
In Exercises 83–88, let logb 2 = A and logb 3 = C and Write each expression in terms of A and C.
logb 8
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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 87
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log x+log(x+3)=log 10
Verified step by step guidance1
Recall the logarithmic property that states \( \log a + \log b = \log (a \times b) \). Use this to combine the left side of the equation: \( \log x + \log (x+3) = \log [x(x+3)] \).
Rewrite the equation using the combined logarithm: \( \log [x(x+3)] = \log 10 \).
Since the logarithms are equal and have the same base, set their arguments equal: \( x(x+3) = 10 \).
Expand and rewrite the equation as a quadratic: \( x^2 + 3x = 10 \), then bring all terms to one side to get \( x^2 + 3x - 10 = 0 \).
Solve the quadratic equation using the quadratic formula or factoring, then check each solution to ensure it makes the arguments of the original logarithms positive (i.e., \( x > 0 \) and \( x+3 > 0 \)) to determine valid solutions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Understanding the properties of logarithms, especially the product rule, is essential. The product rule states that log(a) + log(b) = log(ab), which allows combining multiple logarithmic terms into a single logarithm, simplifying the equation for easier solving.
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Change of Base Property
Domain of Logarithmic Functions
The domain of a logarithmic function includes only positive real numbers. When solving logarithmic equations, it is crucial to check that the solutions do not make any argument of the logarithm zero or negative, as these values are not valid in the original equation.
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Solving Logarithmic Equations
After applying logarithmic properties, the equation often reduces to an algebraic form. Solving this algebraic equation involves isolating the variable and finding exact solutions, which can then be approximated using a calculator if needed.
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5:02Solving Logarithmic Equations
Related Practice
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