Textbook Question
Evaluate each expression without using a calculator. log2 (1/8)
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 27
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb ((x2y)/z2)
Verified step by step guidance1
Recall the logarithmic property for a quotient: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). Apply this to the given expression to separate the numerator and denominator inside the logarithm.
Use the logarithmic property for a product: \(\log_b (MN) = \log_b M + \log_b N\). Apply this to the numerator \(x^2 y\) to split it into two separate logarithms.
Apply the power rule of logarithms: \(\log_b (M^k) = k \log_b M\). Use this to bring down the exponents in the terms \(x^2\) and \(z^2\).
Rewrite the expression by combining all the steps: express the original logarithm as a sum and difference of logarithms with coefficients representing the exponents.
Check if any terms can be simplified further or evaluated without a calculator, such as logarithms of 1 or other known values.
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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 the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^k) = k·log_b(M). These rules are essential for breaking down complex expressions into simpler parts.
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Logarithmic Expansion
Logarithmic expansion involves rewriting a logarithm of a product, quotient, or power as a sum, difference, or multiple of logarithms. This process helps in simplifying expressions and solving equations by expressing the logarithm of a complex expression in terms of simpler logarithms.
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Evaluating Logarithms Without a Calculator
Evaluating logarithms without a calculator often requires recognizing perfect powers or using known logarithm values. By applying logarithmic properties to simplify expressions, one can sometimes reduce the problem to basic logarithms that are easier to evaluate mentally or by using known values.
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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. 5x=17
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
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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. 5ex=23
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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