Question

Find the largest value of x that satisfies:log3(x2)−log3(x+3)=5 x= You may enter the exact value or round to 4 decimal places.

Answer 1

Given:

$\log _3\left(x^2\right)-\log _3\left(x+3\right)=5$

We have:

$\log _3\left(x^2\right)-\log _3\left(x+3\right)+\log _3\left(x+3\right)=5+\log _3\left(x+3\right)$

Simplify:

$\log _3\left(x^2\right)=5+\log _3\left(x+3\right)$

Apply the properties of logarithms:

$x^2=243\left(x+3\right)$

Simplify:

$\begin{gathered} x^2=243x+729 \\ x^2-243x-729=0 \end{gathered}$

We solve using the general formula for quadratic equations, where:

a = 1

b = - 243

c = - 729

So:

$\begin{gathered} x=\frac{-(-243)\pm\sqrt{(-243)^2-4(1)(-729)}}{2(1)} \\ Simplify \\ x=\frac{243\pm\sqrt{61965}}{2}=\frac{243\pm27\sqrt{85}}{2} \end{gathered}$

Separate the solutions:

$\begin{gathered} x=\frac{243+27\sqrt{85}}{2}=245.9639 \\ and \\ x=\frac{243-27\sqrt{85}}{2}=-2.9639 \end{gathered}$

Therefore, the largest value of x is 245.9639

Answer: x = 245.9639