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. 2x=64
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 1
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5 (7 × 3)
Verified step by step guidance1
Recall the logarithmic property that states \( \log_b (MN) = \log_b M + \log_b N \). This means the logarithm of a product can be expressed as the sum of the logarithms.
Apply this property to the given expression \( \log_5 (7 \times 3) \). Rewrite it as \( \log_5 7 + \log_5 3 \).
Check if either \( \log_5 7 \) or \( \log_5 3 \) can be simplified further. Since 7 and 3 are not powers of 5, these logarithms cannot be simplified without a calculator.
Therefore, the expanded form of the expression is \( \log_5 7 + \log_5 3 \), which is the fully expanded form using properties of logarithms.
If needed, you can leave the answer in this expanded form or use a calculator to approximate the values, but the problem asks to expand 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. The product rule states that log_b(M × N) = log_b(M) + log_b(N), allowing the expansion of logarithmic expressions involving multiplication into sums of logs.
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Change of Base Property
Change of Base and Evaluation
Evaluating logarithms without a calculator often involves expressing numbers as powers of the base or using known logarithmic values. Understanding how to rewrite expressions helps simplify or approximate values when direct calculation is not feasible.
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Logarithmic Expression Expansion
Expanding logarithmic expressions means rewriting them using logarithm properties to break down complex arguments into simpler parts. This process aids in simplification, solving equations, or further manipulation in algebraic contexts.
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