교과서 질문
In Exercises 60–63, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. (ln x)(ln 1) = 0
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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
5장, 문제 59
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log4(3x+2)=3
검증된 단계별 안내1
Recall the definition of logarithm: if \(\log_{a}(b) = c\), then it is equivalent to the exponential form \(a^{c} = b\).
Rewrite the given equation \(\log_{4}(3x + 2) = 3\) in exponential form: \(4^{3} = 3x + 2\).
Calculate \$4^{3}$ (but do not finalize the numeric value) and set up the equation: \(64 = 3x + 2\).
Solve for \(x\) by isolating it: subtract 2 from both sides to get \(64 - 2 = 3x\), then divide both sides by 3 to find \(x = \frac{62}{3}\).
Check the domain restriction: the argument of the logarithm, \(3x + 2\), must be greater than 0. Substitute \(x = \frac{62}{3}\) to verify \(3\left(\frac{62}{3}\right) + 2 > 0\) to ensure the solution is valid.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Properties of Logarithms
Understanding the properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable. Recognizing how to manipulate logs helps simplify and solve the given equation.
추천 영상:
5:36
Change of Base Property
Domain of Logarithmic Functions
The domain of a logarithmic function includes all values for which the argument (inside the log) is positive. For log_4(3x+2), the expression 3x+2 must be greater than zero. Identifying and applying domain restrictions ensures that solutions are valid and prevents extraneous answers.
추천 영상:
5:26Graphs of Logarithmic Functions
Exact and Approximate Solutions
After solving the equation exactly, it is often necessary to provide a decimal approximation. Using a calculator to find decimal values correct to two decimal places helps interpret the solution practically. Distinguishing between exact and approximate answers is important in many algebra problems.
추천 영상:
4:20Graph Hyperbolas at the Origin
관련 실천
교과서 질문
Give the equation of each exponential function whose graph is shown.
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교과서 질문
Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 4 ln (x + 6) - 3 ln x
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교과서 질문
The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = log x − 1
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교과서 질문
In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log4 0.863
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교과서 질문
The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = log(x − 1)
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