교과서 질문
Give the equation of each exponential function whose graph is shown.
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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 64
In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2^(4x-2) = 64
검증된 단계별 안내1
Rewrite the equation in exponential form: \(2^{4x-2} = 64\). Recognize that 64 can be expressed as a power of 2, specifically \(64 = 2^6\). This gives \(2^{4x-2} = 2^6\).
Since the bases are the same, set the exponents equal to each other: \(4x - 2 = 6\).
Solve for \(x\) by isolating it. First, add 2 to both sides of the equation: \(4x = 8\).
Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{8}{4}\).
Simplify the fraction to find \(x\). If needed, verify the solution by substituting \(x\) back into the original equation.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Exponential Equations
Exponential equations are mathematical expressions in which a variable appears in the exponent. To solve these equations, one often needs to rewrite them in a form that allows for easier manipulation, such as expressing both sides with the same base or using logarithms. Understanding the properties of exponents is crucial for simplifying and solving these types of equations.
추천 영상:
5:47
Solving Exponential Equations Using Logs
Logarithms
Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in an exponential equation. The logarithm of a number is the exponent to which a base must be raised to produce that number. Familiarity with natural logarithms (base e) and common logarithms (base 10) is essential, especially when expressing solutions in logarithmic form.
추천 영상:
7:30Logarithms Introduction
Calculator Use for Approximations
Using a calculator to obtain decimal approximations is often necessary when dealing with logarithmic solutions or complex exponential equations. Understanding how to input logarithmic functions and interpret the results is vital for accurately finding and rounding solutions to the required decimal places, ensuring precision in mathematical communication.
추천 영상:
5:47Solving Exponential Equations Using Logs
관련 실천
교과서 질문
The figure shows the graph of f(x) = ln x. In Exercises 65–74, 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) = ln (x+2)
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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.
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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) = 1-log x
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교과서 질문
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of 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.
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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. (1/2)(log x + log y)
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