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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 79
Chapter 5, Problem 79

In Exercises 74–79, solve each logarithmic equation. log4 (2x+1) = log4 (x-3) + log4 (x+5)

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Recall the logarithmic property that allows you to combine the sum of logarithms with the same base: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to the right side of the equation to combine the two logarithms into one.
Rewrite the equation using the property: \(\log_4 (2x + 1) = \log_4 ((x - 3)(x + 5))\).
Since the logarithms on both sides have the same base and are equal, set their arguments equal to each other: \(2x + 1 = (x - 3)(x + 5)\).
Expand the right side by multiplying the binomials: \((x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15\).
Set up the equation \(2x + 1 = x^2 + 2x - 15\) and then rearrange all terms to one side to form a quadratic equation: \(0 = x^2 + 2x - 15 - 2x - 1\), which simplifies to \(0 = x^2 - 16\). From here, solve for \(x\).

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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, such as the product, quotient, and power rules, is essential. In this problem, the product rule allows combining the sum of logarithms on one side into a single logarithm, simplifying the equation for easier solving.
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Change of Base Property

Solving Logarithmic Equations

Solving logarithmic equations often involves rewriting the equation using logarithm properties and then converting it to an exponential form or equating the arguments of the logarithms when the bases are the same. This approach helps isolate the variable and find its value.
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Solving Logarithmic Equations

Domain Restrictions of Logarithmic Functions

Logarithmic functions are only defined for positive arguments. When solving equations, it is crucial to check that the solutions satisfy the domain restrictions (arguments inside the logs must be greater than zero) to ensure valid answers.
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Related Practice
Textbook Question

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+4)=log x+log 4

Textbook Question

Find the domain of each logarithmic function. f(x) = ln (x-2)²

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Textbook Question

Use a graphing utility and the change-of-base property to graph each function. y = log3 x

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Textbook Question

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(3x−3)=log(x+1)+log 4

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Textbook Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. logπ 63

Textbook Question

Find the domain of each logarithmic function. f(x) = log (2 - x)

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