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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 7
Chapter 5, Problem 7

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7/x)

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1
Recall the logarithmic property that states \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This allows us to rewrite the logarithm of a quotient as the difference of two logarithms.
Apply this property to the given expression \( \log_7 \left( \frac{7}{x} \right) \), which becomes \( \log_7 7 - \log_7 x \).
Recognize that \( \log_7 7 \) asks the question: "To what power must 7 be raised to get 7?" Since 7 to the power of 1 is 7, \( \log_7 7 = 1 \).
Substitute this value back into the expression to get \( 1 - \log_7 x \).
The expression is now fully expanded using logarithmic properties and simplified as much as possible without a calculator.

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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 in simpler or expanded forms. For example, the quotient rule states that log_b(M/N) = log_b(M) - log_b(N), which is essential for expanding expressions like log_7(7/x).
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Change of Base Property

Logarithm of the Base

The logarithm of a number to its own base is always 1, i.e., log_b(b) = 1. This fact helps simplify expressions where the argument of the logarithm is the base itself, such as log_7(7), which equals 1. Recognizing this can simplify parts of the expression without a calculator.
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Logarithms Introduction

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator involves using known values and properties to simplify expressions. For example, recognizing that log_7(7) = 1 and applying the quotient rule can reduce the expression to a simpler form. This skill is important for exact answers in algebraic contexts.
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Related Practice
Textbook Question

In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ex and g(x) = 2ex/2

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