To determine which statement about the natural logarithmic function $y = \ln(x)$ is false, let's analyze each option one by one:

Analyzing Each Statement:

A) The range of $y = \ln(x)$ goes from $-\infty$ to 100.

• The range of $y = \ln(x)$ is actually from $-\infty$ to $+\infty$. There is no upper bound of 100; this statement is incorrect.

B) The function $\ln(x)$ is increasing wherever it is defined.

• The natural logarithm function $\ln(x)$ is an increasing function for all $x > 0$. This statement is correct.

C) The function $\ln(x)$ is concave down wherever it is defined.

• The function $\ln(x)$ is concave down because its second derivative is negative ($\frac{d^2}{dx^2} \ln(x) < 0$). This statement is correct.

D) The domain for $\ln(x)$ is $x > 0$.

• The natural logarithm is defined only for positive values of $x$, so the domain is $x > 0$. This statement is correct.

E) $\ln(e^4) = 4$

• This is true because $\ln(e^4) = 4$ follows directly from the logarithmic property $\ln(e^a) = a$. This statement is correct.

Conclusion:

The false statement is:

A) The range of $y = \ln(x)$ goes from $-\infty$ to 100.

Would you like more details on any of these points or have any questions?

Here are some related questions to explore further:

1. What is the derivative of the natural logarithmic function $y = \ln(x)$?
2. How do you determine the concavity of a function using the second derivative?
3. Can the range of a function ever be limited, and if so, under what conditions?
4. What are some other key properties of logarithmic functions?
5. How do transformations affect the domain and range of logarithmic functions?

Tip: Remember that understanding the basic properties of logarithmic functions, such as their domain, range, and concavity, helps in identifying any erroneous statements quickly.