1. Rewrite the given expression in simpler form

We have

Using the identity $3^{\log_3 A}=A,$ we get

Also,

So the expression becomes

Let

Then we need the 9th term in the expansion of $(a+b)^{10}$ in increasing powers of $b$.

2. Find the 9th term

In the expansion of $(a+b)^{10}$ in increasing powers of $b$, the general term is

The 9th term corresponds to $r=8.$ Thus,

Now, $\binom{10}{8}=45.$ So,

3. Substitute $a$ and $b$

We have $a^2=25^{x-1}+7.$

Also,

Hence,

Since $25^{x-1}=(5^{x-1})^2$, let $y=5^{x-1}.$ Then $25^{x-1}=y^2.$ So,

We are given that this equals $180$:

Divide by $45$:

Therefore,

So,

Factorizing:

Thus,

Recall that $y=5^{x-1}.$ So we get:

• If $5^{x-1}=1$, then $x-1=0$, hence $x=1.$
• If $5^{x-1}=3$, then $x=1+\log_5 3,$ which is not among the options.

Thus, the possible value from the given options is $\boxed{1}.$

4. Check options

• A: $x=0$ gives $5^{x-1}=5^{-1}=\frac15$, not a solution.
• B: $x=-1$ gives $5^{x-1}=5^{-2}=\frac1{25}$, not a solution.
• C: $x=2$ gives $5^{x-1}=5$, not a solution.
• D: $x=1$ works.

So the correct option is D.

5. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

They agree.