We have the equation:

$(\log_5(x))^2 - 3\log_5(x) + 2 = 0$

This is a quadratic equation in terms of $\log_5(x)$. Let's introduce a substitution to simplify the problem:

Let $y = \log_5(x)$. Then, the equation becomes:

$y^2 - 3y + 2 = 0$

Step 1: Solve the quadratic equation

We can factor the quadratic equation:

$y^2 - 3y + 2 = (y - 1)(y - 2) = 0$

Thus, we have two possible solutions for $y$:

$y - 1 = 0 \quad \text{or} \quad y - 2 = 0$

So, $y = 1$ or $y = 2$.

Step 2: Solve for $x$

Recall that $y = \log_5(x)$. So, we have:

1. If $y = 1$, then $\log_5(x) = 1$, which means $x = 5^1 = 5$.
2. If $y = 2$, then $\log_5(x) = 2$, which means $x = 5^2 = 25$.

Final Answer:

The possible values of $x$ are:

$x = 5 \quad \text{or} \quad x = 25$

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Would you like a more detailed explanation of any step, or have other questions?

Here are 5 related questions:

1. How do logarithmic properties apply to solving equations like this?
2. What is the general approach for solving quadratic equations?
3. Can you explain how logarithms change when the base changes?
4. How do we use logarithmic identities to simplify equations?
5. What is the domain of a logarithmic function, and how does it affect possible solutions?

Tip: When solving logarithmic equations, it's often helpful to introduce a substitution for the logarithm to turn it into a simpler algebraic equation, as done here.