To find the $x$-intercept of the function $f(x) = \log_{2}(-x) - 4$, you set $f(x) = 0$ and solve for $x$. Here's how:

1. Set $f(x) = 0$: $0 = \log_{2}(-x) - 4$

2. Isolate the logarithmic term: $\log_{2}(-x) = 4$

3. Rewrite the logarithmic equation in exponential form: Using the property of logarithms, $\log_{b}(y) = c \implies y = b^c$, $-x = 2^4$

4. Simplify the exponential expression: $-x = 16$

5. Solve for $x$: $x = -16$

$x$-intercept:

The $x$-intercept of the function is at: $x = -16$

Domain Note:

The logarithmic function $\log_{2}(-x)$ is only defined for $x < 0$, so the solution $x = -16$ is valid within the domain of the function.

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Would you like me to graph this function to verify the intercept visually or provide a step-by-step breakdown of logarithmic properties?

Related Questions:

1. How do logarithmic transformations affect the graph of a function?
2. Can this method be generalized for any base of logarithm?
3. What are the domain and range of $f(x) = \log_{2}(-x) - 4$?
4. How do we solve logarithmic equations with different bases?
5. What happens if we set $f(x) = c$ for any $c \neq 0$?

Tip:

Always check the domain restrictions when solving logarithmic functions, as the arguments must be positive for real values.