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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 25
Chapter 5, Problem 25

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log6 (36/(√(x+1))

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Identify the logarithmic expression to expand: \(\log_6 \left( \frac{36}{\sqrt{x+1}} \right)\).
Use the logarithm property for division: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). So rewrite the expression as \(\log_6 36 - \log_6 \sqrt{x+1}\).
Recognize that \(\sqrt{x+1}\) can be written as \((x+1)^{1/2}\), so apply the power rule for logarithms: \(\log_b (M^p) = p \log_b M\). This gives \(\log_6 36 - \frac{1}{2} \log_6 (x+1)\).
Evaluate \(\log_6 36\) by expressing 36 as a power of 6 if possible. Since \(36 = 6^2\), rewrite \(\log_6 36\) as \(\log_6 (6^2)\).
Use the power rule again to simplify \(\log_6 (6^2)\) to \(2 \log_6 6\), and since \(\log_6 6 = 1\), this simplifies to 2. So the expanded form is \(2 - \frac{1}{2} \log_6 (x+1)\).

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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, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^p) = p * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.
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Simplifying Radicals and Exponents

Understanding how to express radicals as fractional exponents is crucial. For instance, the square root of (x+1) can be written as (x+1)^(1/2). This allows the use of the power rule of logarithms to bring the exponent in front of the logarithm, facilitating expansion and simplification.
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Evaluating Logarithms Without a Calculator

Some logarithmic values can be simplified or evaluated exactly by recognizing perfect powers or using known logarithm values. For example, log_6(36) can be simplified since 36 = 6^2, so log_6(36) = 2. This helps in reducing expressions to simpler forms without a calculator.
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Related Practice
Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10x=3.91

Textbook Question

Evaluate each expression without using a calculator. log3 27

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Textbook Question

Evaluate each expression without using a calculator. log5 (1/5)

Textbook Question

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e5

Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2x+1

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ex=5.7