Domain:

\(y\) is a logarithmic function, therefore its natural domain contains those numbers that make the inside positive.

The natural domain is \((-\infty , 5)\).

Continuity:

\(y\) is a logarithmic function, therefore it is continuous on its domain and has no discontinuities.

\(y\) is a logarithmic function, therefore it has a singularity, when the inside is \(0\), which is when \(k = 5\).

Behavior Near a Singularity:

\(\blacktriangleright \) \(y\) is a logarithmic function. The leading coefficient is \(-2\), which is negative. When the domain gets near the singularity \(ln(-k+5)\) becomes unbounded negatively. Since the leading coefficient is negative, we have

End-Behavior:

\(\blacktriangleright \) As \(x\) becomes big and negative, then the inside becomes big and positive. \(ln(big positive)\) is big and positive. Since the leading coefficient is negative, we have

Behavior:

\(y\) is a logarithm function, therefore, is either increasing or decreasing. The leading coefficient is \(-2\), which is negative.

Logarithmic functions have no global or local maximums or minimums. Since the extreme behavior is \(\infty \) to \(-\infty \), we know that \(y\) is a decreasing function.

Zeros:

Since we have a continuous logarithmic function with negative and positive values, it will have a single zero.

The zero will occur when

This all agrees with the graph.