Solve : `log_(x^(2)16+log_(2x)64=3`.

Explanation

To solve the equation \( \log_{x^2} 16 + \log_{2x} 64 = 3 \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of base 2
We know that:
- \( 16 = 2^4 \)
- \( 64 = 2^6 \)

Thus, we can rewrite the equation as:
\[ \log_{x^2} (2^4) + \log_{2x} (2^6) = 3 \]

### Step 2: Apply the logarithmic property
Using the property \( \log_b (a^n) = n \cdot \log_b (a) \), we can simplify:
\[ 4 \cdot \log_{x^2} 2 + 6 \cdot \log_{2x} 2 = 3 \]

### Step 3: Use the change of base formula
We can use the change of base formula \( \log_a b = \frac{\log_c b}{\log_c a} \). Therefore:
\[ \log_{x^2} 2 = \frac{\log_2 2}{\log_2 (x^2)} = \frac{1}{2 \log_2 x} \]
and
\[ \log_{2x} 2 = \frac{\log_2 2}{\log_2 (2x)} = \frac{1}{\log_2 (2) + \log_2 (x)} = \frac{1}{1 + \log_2 x} \]

### Step 4: Substitute back into the equation
Substituting these into our equation gives:
\[ 4 \cdot \frac{1}{2 \log_2 x} + 6 \cdot \frac{1}{1 + \log_2 x} = 3 \]
This simplifies to:
\[ \frac{2}{\log_2 x} + \frac{6}{1 + \log_2 x} = 3 \]

### Step 5: Let \( a = \log_2 x \)
Let \( a = \log_2 x \). Then the equation becomes:
\[ \frac{2}{a} + \frac{6}{1 + a} = 3 \]

### Step 6: Clear the fractions
Multiply through by \( a(1 + a) \) to eliminate the denominators:
\[ 2(1 + a) + 6a = 3a(1 + a) \]
Expanding both sides gives:
\[ 2 + 2a + 6a = 3a + 3a^2 \]
which simplifies to:
\[ 2 + 8a = 3a + 3a^2 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 3a^2 - 5a - 2 = 0 \]

### Step 8: Factor or use the quadratic formula
Now we can factor or use the quadratic formula to solve for \( a \):
\[ (3a + 1)(a - 2) = 0 \]
This gives us:
\[ a = -\frac{1}{3} \quad \text{or} \quad a = 2 \]

### Step 9: Solve for \( x \)
Since \( a = \log_2 x \):
1. For \( a = 2 \):
\[ \log_2 x = 2 \implies x = 2^2 = 4 \]
2. For \( a = -\frac{1}{3} \):
\[ \log_2 x = -\frac{1}{3} \implies x = 2^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{2}} \]

### Final Solution
Thus, the solutions for \( x \) are:
\[ x = 4 \quad \text{and} \quad x = \frac{1}{\sqrt[3]{2}} \]