Given:
[tex]\log _3\left(x^2\right)-\log _3\left(x+3\right)=5[/tex]
We have:
[tex]\log _3\left(x^2\right)-\log _3\left(x+3\right)+\log _3\left(x+3\right)=5+\log _3\left(x+3\right)[/tex]
Simplify:
[tex]\log _3\left(x^2\right)=5+\log _3\left(x+3\right)[/tex]
Apply the properties of logarithms:
[tex]x^2=243\left(x+3\right)[/tex]
Simplify:
[tex]\begin{gathered} x^2=243x+729 \\ x^2-243x-729=0 \end{gathered}[/tex]
We solve using the general formula for quadratic equations, where:
a = 1
b = - 243
c = - 729
So:
[tex]\begin{gathered} x=\frac{-(-243)\pm\sqrt{(-243)^2-4(1)(-729)}}{2(1)} \\ Simplify \\ x=\frac{243\pm\sqrt{61965}}{2}=\frac{243\pm27\sqrt{85}}{2} \end{gathered}[/tex]
Separate the solutions:
[tex]\begin{gathered} x=\frac{243+27\sqrt{85}}{2}=245.9639 \\ and \\ x=\frac{243-27\sqrt{85}}{2}=-2.9639 \end{gathered}[/tex]
Therefore, the largest value of x is 245.9639
Answer: x = 245.9639
