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

Verified step by step guidance

1. 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 \(e^y = x\).
2. Consider the value \(\ln 0\). To find this, we ask: for what value of \(y\) does \(e^y = 0\) hold true?
3. 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.
4. 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.
5. In conclusion, the statement \(\ln 0 = 1\) is false because \(\ln 0\) is not defined in the real numbers.

Key Concepts

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.

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.

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.