Answer :
To determine the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x+3) - 2 \)[/tex], let's break down the problem step by step.
### Step 1: Identify the Vertical Asymptote
The general form of a logarithmic function is [tex]\( \log_b(x - c) + d \)[/tex], where the vertical asymptote occurs at [tex]\( x = c \)[/tex]. For the given function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex]:
[tex]\[ \log(x + 3) \][/tex]
The logarithmic argument [tex]\( x + 3 \)[/tex] must be greater than zero (since the logarithm of a non-positive number is undefined). Therefore, we set up the inequality:
[tex]\[ x + 3 > 0 \][/tex]
[tex]\[ x > -3 \][/tex]
So, the vertical asymptote is at [tex]\( x = -3 \)[/tex], since as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( x + 3 \)[/tex] approaches zero.
### Step 2: Analyze the Behavior Near the Vertical Asymptote
Next, consider the behavior of the function as [tex]\( x \)[/tex] approaches the vertical asymptote [tex]\( -3 \)[/tex] from the right.
For values of [tex]\( x \)[/tex] approaching [tex]\( -3 \)[/tex] from the right (i.e., [tex]\( x > -3 \)[/tex]):
[tex]\[ \log(x + 3) \][/tex]
Since [tex]\( x + 3 \)[/tex] is a small positive number approaching zero, [tex]\( \log(x + 3) \)[/tex] approaches [tex]\( - \infty \)[/tex]. Therefore, as [tex]\( x \)[/tex] nears [tex]\( -3 \)[/tex]:
[tex]\[ \log(x + 3) \rightarrow -\infty \][/tex]
Using this in our function:
[tex]\[ f(x) = \log(x + 3) - 2 \][/tex]
[tex]\[ f(x) \rightarrow -\infty - 2 \][/tex]
[tex]\[ f(x) \rightarrow -\infty \][/tex]
Thus, as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( f(x) \)[/tex] decreases towards [tex]\( -\infty \)[/tex].
### Conclusion
Given these observations, the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] is:
A. As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] decreases to negative infinity.
So, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
### Step 1: Identify the Vertical Asymptote
The general form of a logarithmic function is [tex]\( \log_b(x - c) + d \)[/tex], where the vertical asymptote occurs at [tex]\( x = c \)[/tex]. For the given function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex]:
[tex]\[ \log(x + 3) \][/tex]
The logarithmic argument [tex]\( x + 3 \)[/tex] must be greater than zero (since the logarithm of a non-positive number is undefined). Therefore, we set up the inequality:
[tex]\[ x + 3 > 0 \][/tex]
[tex]\[ x > -3 \][/tex]
So, the vertical asymptote is at [tex]\( x = -3 \)[/tex], since as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( x + 3 \)[/tex] approaches zero.
### Step 2: Analyze the Behavior Near the Vertical Asymptote
Next, consider the behavior of the function as [tex]\( x \)[/tex] approaches the vertical asymptote [tex]\( -3 \)[/tex] from the right.
For values of [tex]\( x \)[/tex] approaching [tex]\( -3 \)[/tex] from the right (i.e., [tex]\( x > -3 \)[/tex]):
[tex]\[ \log(x + 3) \][/tex]
Since [tex]\( x + 3 \)[/tex] is a small positive number approaching zero, [tex]\( \log(x + 3) \)[/tex] approaches [tex]\( - \infty \)[/tex]. Therefore, as [tex]\( x \)[/tex] nears [tex]\( -3 \)[/tex]:
[tex]\[ \log(x + 3) \rightarrow -\infty \][/tex]
Using this in our function:
[tex]\[ f(x) = \log(x + 3) - 2 \][/tex]
[tex]\[ f(x) \rightarrow -\infty - 2 \][/tex]
[tex]\[ f(x) \rightarrow -\infty \][/tex]
Thus, as [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right, [tex]\( f(x) \)[/tex] decreases towards [tex]\( -\infty \)[/tex].
### Conclusion
Given these observations, the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log(x + 3) - 2 \)[/tex] is:
A. As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] decreases to negative infinity.
So, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
