Question:

Step 1: Convert the logarithmic equation using the definition of logarithm. We start with the equation $\log_2(\log_3(x^2)) = 1$. By the definition of logarithm, if $\log_2(y) = 1$, then $y = 2^1 = 2$. Therefore: $$\log_3(x^2) = 2$$ Step 2: Solve the resulting logarithmic equation. Now we have $\log_3(x^2) = 2$. Again applying the definition of logarithm, if $\log_3(z) = 2$, then $z = 3^2 = 9$. Therefore: $$x^2 = 9$$ Step 3: Solve for all possible values of $x$. From $x^2 = 9$, we get: $$x = 3 \quad \text{or} \quad x = -3$$ Step 4: Verify that both solutions satisfy the domain restrictions. For the original equation $\log_2(\log_3(x^2)) = 1$ to be defined, we need: - $x^2 > 0$ (so that $\log_3(x^2)$ is defined) - $\log_3(x^2) > 0$ (so that $\log_2(\log_3(x^2))$ is defined) For $x = 3$: We have $x^2 = 9 > 0$ ✓ and $\log_3(9) = 2 > 0$ ✓ For $x = -3$: We have $x^2 = 9 > 0$ ✓ and $\log_3(9) = 2 > 0$ ✓ Both values satisfy all domain restrictions. Step 5: State the final answer. The equation $\log_2(\log_3(x^2)) = 1$ has **2 solutions**: $x = 3$ and $x = -3$. The answer is **Option 2: 2**