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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 1 - Functions문제 1.55
1장, 문제 1.55

Solving equations Solve the following equations.


ln x= -1

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Step 1: Understand the equation. The equation given is \( \ln x = -1 \). This means we are looking for a value of \( x \) such that the natural logarithm of \( x \) equals \(-1\).
Step 2: Recall the definition of the natural logarithm. The natural logarithm \( \ln x \) is the power to which \( e \) (Euler's number, approximately 2.718) must be raised to obtain \( x \).
Step 3: Convert the logarithmic equation to an exponential form. The equation \( \ln x = -1 \) can be rewritten as \( x = e^{-1} \).
Step 4: Simplify the expression. The expression \( e^{-1} \) can be simplified to \( \frac{1}{e} \).
Step 5: Conclude the solution. The value of \( x \) that satisfies the equation \( \ln x = -1 \) is \( \frac{1}{e} \).

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주요 개념

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

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is the inverse function of the exponential function, meaning that if y = ln(x), then x = e^y. Understanding the properties of natural logarithms is essential for solving equations involving ln.
추천 영상:
05:18
Derivative of the Natural Logarithmic Function

Exponential Function

The exponential function is a mathematical function denoted as e^x, which describes growth or decay processes. It is crucial for solving logarithmic equations, as converting a logarithmic equation to its exponential form allows for easier manipulation. For example, if ln(x) = -1, it can be rewritten as x = e^(-1).
추천 영상:
6:13
Exponential Functions

Solving Logarithmic Equations

Solving logarithmic equations involves isolating the variable by converting the logarithmic expression into its exponential form. This process often requires understanding the properties of logarithms, such as the product, quotient, and power rules. In the case of ln(x) = -1, applying the exponential function helps find the value of x.
추천 영상:
5:02
Solving Logarithmic Equations
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교과서 질문

Solving equations Solve the following equations.


5(ˣ³) = 29

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교과서 질문

Graphing functions Sketch a graph of each function.


g(x) = { 4-2x if x ≤ 1 , (x-1)² + 2 if x > 1

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교과서 질문

Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} f(x)=1x2f\(\left\)(x\(\right\))=\(\frac{1}{x^2}\)

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교과서 질문

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

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교과서 질문

Finding all inverses Find all the inverses associated with the following functions, and state their domains.


ƒ(x) = x² -2x + 6

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교과서 질문

Graph the following functions.

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