Let's solve the given equation step by step:

The problem is to solve for $x$ in the equation:

$\log(x^2) = (\log(x))^2$

Step 1: Simplify $\log(x^2)$

Using the logarithmic property $\log(a^b) = b\log(a)$, we rewrite: $\log(x^2) = 2\log(x)$

So, the equation becomes: $2\log(x) = (\log(x))^2$

Step 2: Rearrange the equation

Rewriting it in standard form: $(\log(x))^2 - 2\log(x) = 0$

Factorizing: $\log(x) \big(\log(x) - 2\big) = 0$

Step 3: Solve for $\log(x)$

From the factorized equation, we have two solutions:

1. $\log(x) = 0$
2. $\log(x) - 2 = 0$, which gives $\log(x) = 2$

Step 4: Solve for $x$

For $\log(x) = 0$: $x = 10^0 = 1$

For $\log(x) = 2$: $x = 10^2 = 100$

Final Solution

The two solutions are: $A = 1, \quad B = 100$

Would you like me to clarify any step further?

Here are some additional related questions for practice:

1. How do logarithmic properties simplify complex equations?
2. Can you solve $\log(x^3) = (\log(x))^3$ for $x$?
3. What happens if the logarithm base is not 10 in this problem?
4. How does changing the base of the logarithm affect the solution?
5. Can you rewrite $\log(x)$ using natural logarithms, and solve?

Tip: