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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 67
Chapter 5, Problem 67

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. log5 x+log5(4x−1)=1

Verified step by step guidance
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Recall the logarithmic property that allows you to combine the sum of two logarithms with the same base: \(\log_b A + \log_b B = \log_b (A \times B)\). Apply this to combine the left side: \(\log_5 x + \log_5 (4x - 1) = \log_5 [x(4x - 1)]\).
Rewrite the equation using the combined logarithm: \(\log_5 [x(4x - 1)] = 1\).
Convert the logarithmic equation to its equivalent exponential form. Since \(\log_b M = N\) means \(b^N = M\), rewrite as \(5^1 = x(4x - 1)\).
Simplify the right side and set up the quadratic equation: \(5 = 4x^2 - x\). Rearrange to standard form: \(4x^2 - x - 5 = 0\).
Solve the quadratic equation \(4x^2 - x - 5 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=4\), \(b=-1\), and \(c=-5\). After finding the solutions, check each to ensure they satisfy the domain restrictions of the original logarithmic expressions (i.e., arguments must be positive).

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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 rule log_b(m) + log_b(n) = log_b(mn), is essential for combining and simplifying logarithmic expressions. This allows the equation to be rewritten in a simpler form, facilitating the solving process.
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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive. When solving logarithmic equations, it is crucial to check that all solutions satisfy this condition to avoid extraneous or invalid answers.
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Graphs of Logarithmic Functions

Solving Exponential Equations

After applying logarithmic properties, the equation often converts to an exponential form. Solving this resulting equation involves algebraic techniques such as factoring or using the quadratic formula to find exact solutions.
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Solving Exponential Equations Using Logs
Related Practice
Textbook Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 8x=121438^x = 12143

Textbook Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.

h(x) = ln (2x)

Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of xx 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. lnx+3=1\(\ln\]\sqrt{x+3}\)=1

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 13[2ln(x+5)−lnx−ln(x2−4)]\(\frac{1}{3}\) \(\left\)[ 2 \(\ln\)(x + 5) - \(\ln\) x - \(\ln\) (x^2 - 4) \(\right\)]

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Textbook Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = ln(x/2)

Textbook Question

In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10^x = 7000

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