Show that $\ln (0)=-\infty$
Answer (2)
Recognize that showing ln ( 0 ) = − ∞ means demonstrating that lim x → 0 + ln ( x ) = − ∞ .
Use the fact that ln ( x ) is the inverse of e x , so y = ln ( x ) if and only if x = e y .
Show that as y approaches − ∞ , e y approaches 0 from the positive side, i.e., lim y → − ∞ e y = 0 + .
Conclude that since y = ln ( x ) , as x approaches 0 from the positive side, ln ( x ) approaches − ∞ , thus ln ( 0 ) = − ∞ .
Explanation
Problem Analysis We are asked to show that ln ( 0 ) = − ∞ . This means we need to demonstrate that as x approaches 0 from the positive side, the natural logarithm of x tends towards negative infinity.
Definition of Natural Logarithm Recall that the natural logarithm, denoted as ln ( x ) , is the inverse function of the exponential function e x . In other words, y = ln ( x ) if and only if x = e y .
Understanding the Limit To show that lim x → 0 + ln ( x ) = − ∞ , we need to show that as x approaches 0 from the positive side (i.e., x takes on values like 0.1, 0.01, 0.001, and so on), ln ( x ) becomes increasingly large in the negative direction.
Using the Exponential Function Equivalently, we can show that as y approaches − ∞ , e y approaches 0 from the positive side. That is, we want to show that lim y → − ∞ e y = 0 + .
Behavior of Exponential Function The exponential function e y is always positive for any real number y . As y becomes a large negative number, e y gets closer and closer to 0. For example:
e − 1 ≈ 0.368
e − 10 ≈ 0.000045
e − 100 is an extremely small positive number.
Thus, as y approaches − ∞ , e y approaches 0 from the positive side.
Conclusion Since y = ln ( x ) , this means that as x approaches 0 from the positive side, ln ( x ) approaches − ∞ . Therefore, we can write: x → 0 + lim ln ( x ) = − ∞ This is often informally written as ln ( 0 ) = − ∞ .
Examples
Understanding the behavior of logarithmic functions as they approach zero is crucial in various fields. For instance, in information theory, the entropy of a system is often expressed using logarithms. When dealing with rare events (probabilities approaching zero), the logarithmic function helps quantify the information content or surprise associated with those events. Similarly, in physics, logarithmic scales are used to represent quantities that vary over a wide range, such as sound intensity (decibels) or acidity (pH). Knowing that ln ( 0 ) = − ∞ helps in understanding the limits and interpretations of these scales.
The limit of the natural logarithm as its argument approaches zero from the positive side is lim x → 0 + ln ( x ) = − ∞ , which implies that ln ( 0 ) = − ∞ . This demonstrates how logarithmic functions behave as they near zero, leading to increasingly negative values. Thus, we can safely conclude that ln ( 0 ) = − ∞ .
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