A logarithmic function of the form $$ y=a+b \log (x+2) $$ passes through the points $$ (-1,-18) $$ and $$ (439,3) $$. Find $$ a $$ and $$ b $$, rounding all values to three decimal places, if necessary. Answer Attempt 1 out of 2 $$ a=\square b=\square $$

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To find the values of $$ a $$ and $$ b $$ in the logarithmic function $$ y = a + b \log(x + 2) $$ that pass through the points $$ (-1, -18) $$ and $$ (439, 3) $$, we can substitute the given points into the function and solve the resulting system of equations.

Given points:
1. $$ (-1, -18) $$
2. $$ (439, 3) $$

Substitute the points into the function:
1. For point $$ (-1, -18) $$:
$$ -18 = a + b \log(-1 + 2) $$
$$ -18 = a + b \log(1) $$
$$ -18 = a + b \cdot 0 $$
$$ -18 = a $$

2. For point $$ (439, 3) $$:
$$ 3 = a + b \log(439 + 2) $$
$$ 3 = a + b \log(441) $$

Now, we have a system of equations:
1. $$ -18 = a $$
2. $$ 3 = a + b \log(441) $$

Solve the system of equations to find the values of $$ a $$ and $$ b $$.
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
  $$\left\{ \begin{array}{l}-18=a\3=a+b\log_{10}{\left(441\right)}\end{array}\right.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}-18=a\3=a+2\log_{10}{\left(21\right)}\times b\end{array}\right.$$
- step2: Solve the equation:
  $$\left\{ \begin{array}{l}a=-18\3=a+2\log_{10}{\left(21\right)}\times b\end{array}\right.$$
- step3: Substitute the value of $$a:$$
  $$3=-18+2\log_{10}{\left(21\right)}\times b$$
- step4: Swap the sides:
  $$-18+2\log_{10}{\left(21\right)}\times b=3$$
- step5: Move the constant to the right side:
  $$2\log_{10}{\left(21\right)}\times b=3+18$$
- step6: Add the numbers:
  $$2\log_{10}{\left(21\right)}\times b=21$$
- step7: Divide both sides:
  $$\frac{2\log_{10}{\left(21\right)}\times b}{2\log_{10}{\left(21\right)}}=\frac{21}{2\log_{10}{\left(21\right)}}$$
- step8: Divide the numbers:
  $$b=\frac{21}{2\log_{10}{\left(21\right)}}$$
- step9: Calculate:
  $$\left\{ \begin{array}{l}a=-18\b=\frac{21}{2\log_{10}{\left(21\right)}}\end{array}\right.$$
- step10: Check the solution:
  $$\left\{ \begin{array}{l}a=-18\b=\frac{21}{2\log_{10}{\left(21\right)}}\end{array}\right.$$
- step11: Rewrite:
  $$\left(a,b\right) = \left(-18,\frac{21}{2\log_{10}{\left(21\right)}}\right)$$
The solution to the system of equations is:
$$ a = -18 $$
$$ b = \frac{21}{2\log_{10}(21)} $$

Rounding the values to three decimal places, we have:
$$ a \approx -18 $$
$$ b \approx 7.941 $$

Therefore, the values of $$ a $$ and $$ b $$ are approximately $$ a = -18 $$ and $$ b = 7.941 $$.
$$ a = -18 $$ and $$ b \approx 7.941 $$

A logarithmic function of the form \( y=a+b \log (x+2) \) passes through the points \( (-1,-18) \) and \( (439,3) \). Find \( a \) and \( b \), rounding all values to three decimal places, if necessary. Answer Attempt 1 out of 2 \( a=\square b=\square \)

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