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Solve the equation. Write the solution set with the exact solutions.

[tex]\[ \log \left(q^2 + 7q\right) = \log 18 \][/tex]

If there is more than one solution, separate the answers with commas. If there is no solution, write [tex]\(\{\}\)[/tex].

The exact solution set is [tex]\(\square\)[/tex].


Sagot :

To solve the equation, [tex]\(\log \left(q^2 + 7q\right) = \log 18\)[/tex], follow these steps:

1. Remove the logarithms:
Since we have [tex]\(\log A = \log B\)[/tex], we can equate the arguments:
[tex]\[ q^2 + 7q = 18 \][/tex]

2. Set up the quadratic equation:
Rewrite the equation in standard form:
[tex]\[ q^2 + 7q - 18 = 0 \][/tex]

3. Solve the quadratic equation:
To solve [tex]\(q^2 + 7q - 18 = 0\)[/tex], factorize or use the quadratic formula [tex]\(q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] where [tex]\(a = 1\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = -18\)[/tex].

4. Find the solutions:
Solve the quadratic equation by factoring:
[tex]\[ (q + 9)(q - 2) = 0 \][/tex]

Set each factor to zero and solve for [tex]\(q\)[/tex]:
[tex]\[ q + 9 = 0 \quad \Rightarrow \quad q = -9 \][/tex]
[tex]\[ q - 2 = 0 \quad \Rightarrow \quad q = 2 \][/tex]

5. Verify the solutions:
Substitute [tex]\(q = -9\)[/tex] and [tex]\(q = 2\)[/tex] back into the original argument [tex]\(q^2 + 7q\)[/tex] to ensure the solutions are valid:
[tex]\[ \log((-9)^2 + 7(-9)) = \log(81 - 63) = \log(18) \quad \text{(valid)} \][/tex]
[tex]\[ \log(2^2 + 7 \cdot 2) = \log(4 + 14) = \log(18) \quad \text{(valid)} \][/tex]

Both solutions are valid. Thus, the exact solution set is:
[tex]\[ \boxed{-9, 2} \][/tex]