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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 4 - Exponential and Logarithmic Functions문제 39
5장, 문제 39

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

검증된 단계별 안내
1
Identify the base function f(x) = \(\log\) x, which is the logarithmic function with vertical asymptote at x = 0 and domain (0, \(\infty\)). Its range is all real numbers (-\(\infty\), \(\infty\)).
Analyze the function g(x) = -\(\log\)(x+3). Notice that the argument of the logarithm is (x + 3), which means the graph of f(x) is shifted horizontally to the left by 3 units. This changes the vertical asymptote from x = 0 to x = -3.
The negative sign in front of the logarithm reflects the graph of f(x) across the x-axis. So, after shifting the graph of f(x) left by 3 units, reflect it over the x-axis to get g(x).
Write the equation of the vertical asymptote for g(x) as x = -3, since the logarithm is undefined when its argument is zero, i.e., x + 3 = 0.
Determine the domain and range of g(x): the domain is all x such that x + 3 > 0, or x > -3, and the range remains all real numbers (-\(\infty\), \(\infty\)) because reflection and horizontal shifts do not restrict the range of the logarithmic function.

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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.

Logarithmic Functions and Their Graphs

A logarithmic function, such as f(x) = log x, is the inverse of an exponential function. Its graph passes through (1,0) and has a vertical asymptote at x = 0. The function is defined only for positive x-values, and its range is all real numbers.
추천 영상:
5:26
Graphs of Logarithmic Functions

Transformations of Functions

Transformations include shifts, reflections, and stretches/compressions of a graph. For g(x) = -log(x+3), the graph of f(x) = log x is shifted left by 3 units and reflected across the x-axis. Understanding these helps in sketching g(x) from f(x) and identifying changes in domain and range.
추천 영상:
4:22
Domain & Range of Transformed Functions

Asymptotes and Domain/Range of Logarithmic Functions

Logarithmic functions have vertical asymptotes where the argument equals zero. For f(x) = log x, the asymptote is x = 0; for g(x) = -log(x+3), it is x = -3. The domain excludes these values, while the range remains all real numbers. Recognizing asymptotes is key to understanding function behavior.
추천 영상:
5:26
Graphs of Logarithmic Functions
관련 실천
교과서 질문

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(10x21−x37(x+1)2)\(\log\) \(\left\)( \(\frac{10x^2 \sqrt[3]{1 - x}\)}{7(x + 1)^2} \(\right\))

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교과서 질문

Evaluate each expression without using a calculator. log5 57

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교과서 질문

Evaluate each expression without using a calculator. 8log8198^{\(\log\)_8 19}

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교과서 질문

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)

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교과서 질문

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. 70.3x=813

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교과서 질문

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. 5(2x+3)=3(x−1)

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