Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 42x−1=64
747
views
Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
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)
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
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).
Recommended video:
5:36
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.
Recommended video:
7:30Logarithms 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.
Recommended video:
5:14Evaluate Logarithms
Related Practice
Textbook Question
Write each equation in its equivalent exponential form. 5= logb 32
880
views
Textbook Question
In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. e2.3
1050
views
Textbook Question
In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. 4-1.5
794
views
Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 22x-1=32
766
views
Textbook Question
Write each equation in its equivalent exponential form. log6 216 = y
839
views