which of the following statements about the natural logarithmic function y=In(x) is FALSA? A) the range of Y=In(X) goes from - infinity to 100.  b) the function In(x) is increasing wherever it is defined.  c) the function In(x) is concave down wherever it is defined.   d) the domain for In(x) is x>0.   e) In(e^4)=4

Let's analyze each statement carefully:

A) **The range of \( y = \ln(x) \) goes from -infinity to 100.**  
- The actual range of the natural logarithmic function \( y = \ln(x) \) is from **-infinity to infinity**, not just up to 100. So, this statement is **false**.

B) **The function \( \ln(x) \) is increasing wherever it is defined.**  
- The derivative of \( \ln(x) \) is \( \frac{d}{dx} \ln(x) = \frac{1}{x} \), which is positive for \( x > 0 \). Therefore, the function is indeed increasing wherever it is defined. This statement is **true**.

C) **The function \( \ln(x) \) is concave down wherever it is defined.**  
- The second derivative of \( \ln(x) \) is \( \frac{d^2}{dx^2} \ln(x) = -\frac{1}{x^2} \), which is negative for \( x > 0 \). This confirms that \( \ln(x) \) is concave down wherever it is defined. This statement is **true**.

D) **The domain for \( \ln(x) \) is \( x > 0 \).**  
- The natural logarithmic function is only defined for positive values of \( x \). This statement is **true**.

E) **\( \ln(e^4) = 4 \).**  
- Using the logarithmic property \( \ln(e^a) = a \), we can see that \( \ln(e^4) = 4 \). This statement is **true**.

### Conclusion:
The false statement is **A)**. The range of \( y = \ln(x) \) is from -infinity to infinity, not just up to 100.

Would you like any more details on this?

Here are 5 related questions to consider:
1. What is the range of the natural logarithmic function \( \ln(x) \)?
2. Why is \( \ln(x) \) concave down based on its second derivative?
3. How do logarithmic properties simplify expressions like \( \ln(e^x) \)?
4. What is the general domain of logarithmic functions?
5. How can you prove that \( \ln(x) \) is an increasing function using derivatives?

**Tip:** When solving logarithmic problems, remembering key properties like \( \ln(e^

Which of the following statements about the natural logarithmic function y=In(x) is FALSA? A) the range of Y=In(X) goes from - infinity to 100.  B) the function In(x) is increasing wherever it is defined.  C) the function In(x) is concave down wherever it is defined.  D) the domain for In(x) is x>0.  E) In(e^4)=4

Discover how to analyze the natural logarithmic function y = ln(x) by evaluating true and false statements about its range, domain, concavity, and properties. This guide breaks down key concepts like the derivative of ln(x), concavity rules, and logarithmic identities, suitable for Grades 11-12 or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation