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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 53
5장, 문제 53

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(x+5)=3

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1
Identify the given logarithmic equation: \(\log_{4}(x+5) = 3\).
Recall the definition of logarithm: \(\log_{a}(b) = c\) means \(a^{c} = b\). Apply this to rewrite the equation as \(4^{3} = x + 5\).
Calculate the value of \$4^{3}\( (but do not finalize the numeric value here), so the equation becomes \)x + 5 = 4^{3}$.
Isolate \(x\) by subtracting 5 from both sides: \(x = 4^{3} - 5\).
Check the domain restriction: since the argument of the logarithm \(x + 5\) must be greater than 0, ensure that \(x + 5 > 0\), which means \(x > -5\). Confirm that your solution satisfies this domain condition.

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주요 개념

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

Definition and Properties of Logarithms

A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this definition allows you to rewrite logarithmic equations in exponential form, which is essential for solving equations like log_4(x+5) = 3.
추천 영상:
5:36
Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(f(x)) requires that the argument f(x) be positive. This means x+5 > 0 in the given problem, so x must be greater than -5. Checking the domain ensures that any solution found is valid within the original equation.
추천 영상:
5:26
Graphs of Logarithmic Functions

Converting Logarithmic Equations to Exponential Form

To solve log_b(f(x)) = c, rewrite it as f(x) = b^c. For log_4(x+5) = 3, this becomes x+5 = 4^3. This conversion simplifies the problem to solving an algebraic equation, making it easier to find the exact value of x.
추천 영상:
5:02
Solving Logarithmic Equations
관련 실천
교과서 질문

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log₂ (x + 1)

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교과서 질문

Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = (½)x and g(x) = (½)x-1 + 1

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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. 2 logb x + 3 logb y

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교과서 질문

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. ln x=2

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교과서 질문

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. lnxe3\(\ln\]\sqrt\)[3]{\(\frac{x}{e}\)}

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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)ln x + ln y

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