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

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

검증된 단계별 안내
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Start with the given exponential equation: \(7^{0.3x} = 813\).
To solve for \(x\), take the natural logarithm (or common logarithm) of both sides to utilize the property that \(\ln(a^b) = b \ln(a)\): \(\ln(7^{0.3x}) = \ln(813)\).
Apply the logarithm power rule to bring down the exponent: \(0.3x \cdot \ln(7) = \ln(813)\).
Isolate \(x\) by dividing both sides by \(0.3 \ln(7)\): \(x = \frac{\ln(813)}{0.3 \cdot \ln(7)}\).
Use a calculator to find the decimal values of the logarithms and compute the quotient to get the approximate value of \(x\) rounded to two decimal places.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Equations

An exponential equation is one in which the variable appears in the exponent. Solving such equations often involves rewriting the equation to isolate the exponential expression and then applying logarithms to both sides to solve for the variable.
추천 영상:
5:47
Solving Exponential Equations Using Logs

Logarithms and Their Properties

Logarithms are the inverse operations of exponentials. They allow us to solve equations where the variable is an exponent by converting the exponential form into a logarithmic form. Common logarithms (base 10) and natural logarithms (base e) are frequently used.
추천 영상:
5:36
Change of Base Property

Using Calculators for Approximation

After expressing the solution in logarithmic form, calculators are used to find decimal approximations. This step involves evaluating logarithmic expressions and rounding the result to the desired decimal places, ensuring practical and usable answers.
추천 영상:
5:47
Solving Exponential Equations Using Logs
관련 실천
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. 7(x+2)=410

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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) = log x and g(x) = - log (x+3)

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교과서 질문

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = ex-1+2

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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(10x21x37(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

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = ex+2

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