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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.1.67b
7장, 문제 7.1.67b

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


b. ln 0 = 1

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Recall the definition of the natural logarithm function: \(\ln x\) is the inverse of the exponential function \(e^x\), meaning \(\ln x = y\) if and only if \(e^y = x\).
Consider the value \(\ln 0\). To find this, we ask: for what value of \(y\) does \(e^y = 0\) hold true?
Since the exponential function \(e^y\) is always positive for all real numbers \(y\) (i.e., \(e^y > 0\) for all \(y\)), it never equals zero.
Therefore, there is no real number \(y\) such that \(e^y = 0\), which means \(\ln 0\) is undefined and does not equal 1 or any other real number.
In conclusion, the statement \(\ln 0 = 1\) is false because \(\ln 0\) is not defined in the real numbers.

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Definition and Domain of the Natural Logarithm Function

The natural logarithm function, ln(x), is defined only for positive real numbers (x > 0). It represents the inverse of the exponential function e^x, meaning ln(x) answers the question: 'To what power must e be raised to get x?' Since 0 is not positive, ln(0) is undefined.
추천 영상:
05:18
Derivative of the Natural Logarithmic Function

Behavior of ln(x) as x Approaches Zero

As x approaches 0 from the positive side, ln(x) decreases without bound, tending toward negative infinity. This means ln(0) is not a finite number and certainly not equal to 1. Understanding this limit helps clarify why ln(0) is undefined.
추천 영상:
05:36
Integrals of Natural Exponential Functions (e^x) Example 3

Evaluating the Truth of Mathematical Statements

To determine if a statement like 'ln 0 = 1' is true, one must check the domain and properties of the functions involved. Since ln(0) is undefined, the statement is false. Providing a counterexample or referencing the function's domain is essential in justifying such claims.
추천 영상:
5:14
Evaluate Logarithms
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61–62. Points of intersection and area

b. Compute the area of the region described.


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b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t ≥ 0.

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b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?

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