Let's solve the given logarithmic equation step-by-step:
[tex]\[ \log (\sqrt{x+4}) - \log (3 x) = -2 \log (3) \][/tex]
First, we'll simplify the logarithmic expressions using logarithmic properties.
1. Logarithmic Properties: Recall a key logarithmic property:
[tex]\[ \log(a) - \log(b) = \log \left(\frac{a}{b}\right) \][/tex]
Applying this property to our equation:
[tex]\[ \log (\sqrt{x+4}) - \log (3 x) = \log \left(\frac{\sqrt{x+4}}{3x}\right) \][/tex]
2. Simplifying the Logarithmic Equation: Now rewrite the equation using the property mentioned above:
[tex]\[ \log \left(\frac{\sqrt{x+4}}{3x}\right) = -2 \log (3) \][/tex]
3. Converting Logarithmic Form to Exponential Form: Recognize that
[tex]\[ -2 \log (3) = \log(3^{-2}) = \log \left(\frac{1}{3^2}\right) = \log \left(\frac{1}{9}\right) \][/tex]
Consequently, the equation becomes:
[tex]\[ \log \left(\frac{\sqrt{x+4}}{3x}\right) = \log \left(\frac{1}{9}\right) \][/tex]
4. Eliminating the Logarithms: Since the logarithms are equal, the arguments must be equal as well:
[tex]\[ \frac{\sqrt{x+4}}{3x} = \frac{1}{9} \][/tex]
5. Solving the Resulting Equation: Cross-multiply to remove the fractions:
[tex]\[ 9 \sqrt{x+4} = 3x \][/tex]
6. Isolating the Radical: Divide both sides by 3 to simplify:
[tex]\[ 3 \sqrt{x+4} = x \][/tex]
7. Squaring Both Sides: To eliminate the square root, square both sides of the equation:
[tex]\[ (3 \sqrt{x+4})^2 = x^2 \][/tex]
[tex]\[ 9 (x+4) = x^2 \][/tex]
8. Forming a Quadratic Equation: Expand and simplify:
[tex]\[ 9x + 36 = x^2 \][/tex]
[tex]\[ x^2 - 9x - 36 = 0 \][/tex]
9. Factoring the Quadratic Equation: Solve the quadratic equation:
[tex]\[ x^2 - 9x - 36 = 0 \][/tex]
Using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -9 \)[/tex], and [tex]\( c = -36 \)[/tex].
Substitute these values in:
[tex]\[ x = \frac{9 \pm \sqrt{81 + 144}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{225}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm 15}{2} \][/tex]
This gives us two potential solutions:
[tex]\[ x = \frac{24}{2} = 12 \][/tex]
[tex]\[ x = \frac{-6}{2} = -3 \][/tex]
10. Verifying the Solutions: Since [tex]\( x = -3 \)[/tex] would make the logarithms undefined (logarithms are not defined for non-positive arguments), we discard this solution.
Therefore, the valid solution is:
[tex]\[ x = 12 \][/tex]
So, the solution to the given equation is:
[tex]\[ x = 12 \][/tex]
