The problem is to find the constants $a$ and $b$ in the logarithmic function:

$y = a + b \ln(x - 5)$

where the function passes through the points $(18, 17)$ and $(930, -16)$.

We can set up a system of equations using these two points. For each point, we substitute the corresponding values of $x$ and $y$ into the equation.

Step 1: Write equations for both points

For the point $(18, 17)$: $17 = a + b \ln(18 - 5)$ $17 = a + b \ln(13)$

For the point $(930, -16)$: $-16 = a + b \ln(930 - 5)$ $-16 = a + b \ln(925)$

Now we have the system of two equations: $17 = a + b \ln(13)$ $-16 = a + b \ln(925)$

Step 2: Solve the system of equations

Subtract the first equation from the second:

$(-16) - 17 = \left(a + b \ln(925)\right) - \left(a + b \ln(13)\right)$ $-33 = b \left(\ln(925) - \ln(13)\right)$ $-33 = b \ln\left(\frac{925}{13}\right)$

We calculate: $\frac{925}{13} \approx 71.1538$ $\ln(71.1538) \approx 4.266$

Thus, the equation becomes: $-33 = b \times 4.266$

Solve for $b$: $b = \frac{-33}{4.266} \approx -7.741$

Step 3: Solve for $a$

Now substitute $b = -7.741$ into the first equation: $17 = a + (-7.741) \ln(13)$

We know that $\ln(13) \approx 2.5657$, so: $17 = a - 7.741 \times 2.5657$ $17 = a - 19.877$

Solve for $a$: $a = 17 + 19.877 \approx 36.877$

Final Answer:

The values of $a$ and $b$ are: $a \approx 36.877 \quad \text{and} \quad b \approx -7.741$

Would you like further explanation on any part of the solution or have any questions? Here are some related questions that might interest you:

1. How does the natural logarithm function behave with large values of $x$?
2. What happens if the function were to pass through more than two points?
3. How can you graph a logarithmic function with the parameters found?
4. What is the effect of changing the constant $a$ in the function?
5. How would you find the constants $a$ and $b$ for an exponential function instead?

Tip: When working with logarithmic functions, always check if the argument inside the logarithm is positive since the logarithm of a non-positive number is undefined.