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=(-(-243)\pm√((-243)^2-4(1)(-729)))/(2(1)) \\ Simplify \\ x=(243\pm√(61965))/(2)=(243\pm27√(85))/(2) \end{gathered}`

Separate the solutions:

`\begin{gathered} x=(243+27√(85))/(2)=245.9639 \\ and \\ x=(243-27√(85))/(2)=-2.9639 \end{gathered}`

Therefore, the largest value of x is 245.9639

Answer: x = 245.9639