The function you're describing is:

$f(x) = \log_2(x) - 3$

Let's break down the key features of this function:

1. Domain

The function $f(x) = \log_2(x) - 3$ involves a logarithmic expression. The domain of a logarithmic function is restricted to values of $x$ that make the argument of the logarithm positive. Therefore, for $\log_2(x)$ to be defined, we need:

$x > 0$

Thus, the domain of this function is:

$(0, \infty)$

2. Vertical Asymptote

Logarithmic functions have vertical asymptotes where the argument approaches zero. Since the argument of the logarithm is $x$, there is a vertical asymptote at:

$x = 0$

This means the graph will approach negative infinity as $x$ gets closer to zero from the positive side.

3. Horizontal Shift

The term $-3$ shifts the graph of $\log_2(x)$ downward by 3 units. This means the function will have the same shape as the basic logarithmic graph but will be lowered along the vertical axis by 3 units.

4. Intercept

To find the $x$-intercept (where $f(x) = 0$):

$0 = \log_2(x) - 3$ $\log_2(x) = 3$ $x = 2^3 = 8$

So, the $x$-intercept is at $(8, 0)$.

5. Range

The range of the function is all real numbers, because logarithmic functions can take on any real value as $x$ increases or decreases within the domain. So, the range is:

$(-\infty, \infty)$

6. Behavior as $x$ Approaches Infinity and Zero

• As $x \to \infty$, $\log_2(x) \to \infty$, so the function will increase without bound.
• As $x \to 0^+$, $\log_2(x) \to -\infty$, so $f(x) \to -\infty$, as the vertical asymptote at $x = 0$ is approached.

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Would you like me to go over any of these features in more detail, or do you have any specific questions about the graph?

Here are some related questions you might find interesting:

1. What happens to the graph if the function was $f(x) = \log_2(x + 3)$?
2. How would the graph change if the function were $f(x) = 2 \log_2(x) - 3$?
3. How do vertical asymptotes affect the behavior of logarithmic functions?
4. How can you solve logarithmic equations like $\log_2(x) = 5$?
5. What are the transformations that affect the graph of $f(x) = \log_2(x)$?

Tip: For any logarithmic function, identifying the shifts and vertical asymptotes is key to understanding its graph!