We are given the logarithmic function:

$y = a + b \log (x+2)$

and the points $(-1, -18)$ and $(439, 3)$.
We need to determine $a$ and $b$ by forming a system of equations.

Step 1: Substitute the First Point $(-1, -18)$

$-18 = a + b \log((-1) + 2)$

Since $\log(1) = 0$, this simplifies to: $-18 = a + b(0) \Rightarrow a = -18$

Step 2: Substitute the Second Point $(439, 3)$

$3 = -18 + b \log(439 + 2)$

Since $439 + 2 = 441$, we compute: $3 = -18 + b \log(441)$

Using $\log(441) \approx 2.645$, $3 = -18 + b(2.645)$

Step 3: Solve for $b$

$3 + 18 = b(2.645)$ $21 = b(2.645)$ $b = \frac{21}{2.645} \approx 7.942$

Final Answer:

$a = -18, \quad b \approx 7.942$

Would you like a more detailed breakdown of any step?

Related Questions:

1. What is the general form of a logarithmic function?
2. How do you determine the domain of a logarithmic function?
3. Why does $\log(1) = 0$ simplify the equation?
4. What is the significance of the base in logarithmic functions?
5. How can we check the accuracy of our computed values?

Tip:

When solving for parameters in logarithmic equations, always check if the logarithm argument is positive to avoid undefined values.