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^