Find the largest value of $$ x $$ that satisfies: $$ \log _{3}\left(x^{2}\right)-\log _{3}(x+3)=5 $$ $$ x=\square $$

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UpStudy AI · Accepted Answer

We start with the equation
$$
\log_{3}\left(x^{2}\right)-\log_{3}(x+3)=5.
$$
Using the logarithm rule
$$
\log_b A - \log_b B = \log_b\left(\frac{A}{B}\right),
$$
we combine the logarithms to obtain
$$
\log_{3}\left(\frac{x^2}{x+3}\right)=5.
$$

Next, we rewrite the logarithmic equation in its exponential form:
$$
\frac{x^2}{x+3}=3^5.
$$
Since
$$
3^5=243,
$$
the equation becomes
$$
\frac{x^2}{x+3}=243.
$$

Multiply both sides by $$x+3$$ (noting that the domain requires $$x+3>0$$, i.e. $$x>-3$$ and also $$x\neq0$$ because $$\log_3(x^2)$$ requires $$x^2>0$$):
$$
x^2 = 243(x+3).
$$

Rearrange the equation to form a quadratic:
$$
x^2 - 243x - 729 = 0.
$$

To solve the quadratic $$x^2 - 243x - 729 = 0$$, we use the quadratic formula:
$$
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},
$$
where $$a=1$$, $$b=-243$$, and $$c=-729$$. Compute the discriminant:
$$
b^2-4ac = (-243)^2 - 4(1)(-729)=59049+2916=61965.
$$

Notice that:
$$
61965=729\cdot85 \quad \Longrightarrow \quad \sqrt{61965}=\sqrt{729\cdot85}=27\sqrt{85}.
$$

Thus, the solutions are
$$
x=\frac{243\pm27\sqrt{85}}{2}.
$$

Since we are asked for the largest value of $$ x $$ and considering the domain restrictions, we choose the positive solution:
$$
x=\frac{243+27\sqrt{85}}{2}.
$$
The largest value of $$ x $$ that satisfies the equation is $$ x=\frac{243+27\sqrt{85}}{2} $$.

Find the largest value of that satisfies:

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