To determine the correct statement about the end behavior of the logarithmic function [tex]\( f(x) = \log (x + 3) - 2 \)[/tex], follow these steps:
1. Identify the vertical asymptote:
- The vertical asymptote occurs where the argument of the logarithm [tex]\( (x + 3) \)[/tex] is zero.
- Set [tex]\( x + 3 = 0 \)[/tex], which gives [tex]\( x = -3 \)[/tex].
2. Analyze the behavior of [tex]\( f(x) \)[/tex] near the vertical asymptote [tex]\( x = -3 \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\( -3 \)[/tex] from the right (values of [tex]\( x \)[/tex] greater than [tex]\(-3\)[/tex] but close to [tex]\(-3\)[/tex]), the argument [tex]\( x + 3 \)[/tex] approaches zero from the positive side.
- The logarithmic function [tex]\( \log (x + 3) \)[/tex] tends to [tex]\(-\infty\)[/tex] as [tex]\( x + 3 \)[/tex] gets closer to zero from the positive side, because the natural log of a very small positive number is a large negative number.
- Therefore, [tex]\( \log (x + 3) - 2 \)[/tex] will also tend towards [tex]\(-\infty\)[/tex].
3. Summarize the end behavior:
- As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] (i.e., [tex]\( f(x) \)[/tex]) decreases to negative infinity.
Given the above analysis, the correct statement is:
A. As [tex]\( x \)[/tex] decreases to the vertical asymptote at [tex]\( x = -3 \)[/tex], [tex]\( y \)[/tex] decreases to negative infinity.
