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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

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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Start with the given expression: \(\frac{1}{3} \left[ 2 \ln(x + 5) - \ln x - \ln(x^2 - 4) \right]\).
Use the logarithm power rule to move coefficients inside the logarithms as exponents: \(2 \ln(x + 5) = \ln((x + 5)^2)\).
Rewrite the expression inside the brackets as a single logarithm using the properties of logarithms: \(\ln((x + 5)^2) - \ln x - \ln(x^2 - 4) = \ln \left( \frac{(x + 5)^2}{x (x^2 - 4)} \right)\).
Recall that \(x^2 - 4\) is a difference of squares and can be factored as \((x - 2)(x + 2)\), so rewrite the denominator accordingly: \(x (x - 2)(x + 2)\).
Now apply the outer coefficient \(\frac{1}{3}\) as a power to the entire logarithm: \(\frac{1}{3} \ln \left( \frac{(x + 5)^2}{x (x - 2)(x + 2)} \right) = \ln \left( \left( \frac{(x + 5)^2}{x (x - 2)(x + 2)} \right)^{\frac{1}{3}} \right)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Properties of logarithms include rules such as the product rule, quotient rule, and power rule. These allow combining or breaking down logarithmic expressions: for example, log(a) + log(b) = log(ab), log(a) - log(b) = log(a/b), and k·log(a) = log(a^k). Understanding these is essential to condense multiple logarithms into a single expression.
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Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e (approximately 2.718). It has the same properties as other logarithms but is commonly used in calculus and algebra. Recognizing ln and its behavior helps in simplifying expressions and evaluating them when possible.
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Simplifying Algebraic Expressions Inside Logarithms

Before condensing logarithms, simplify the algebraic expressions inside them, such as factoring or recognizing differences of squares (e.g., x^2 - 4 = (x-2)(x+2)). This simplification aids in combining logarithms correctly and can make evaluation or further manipulation easier.
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Related Practice
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. g(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. 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

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

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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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