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 using properties of logarithms
We can express 16 and 64 in terms of powers of 4:
- \( 16 = 4^2 \)
- \( 64 = 4^3 \)

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

### Step 2: Apply the power rule of logarithms
Using the property \( \log_b (a^n) = n \log_b (a) \), we get:
\[ 2 \log_{x^2} 4 + 3 \log_{2x} 4 = 3 \]

### Step 3: Change the base of the logarithms
Using the change of base formula \( \log_a b = \frac{\log_c b}{\log_c a} \), we can express the logarithms in terms of base 2:
\[ 2 \cdot \frac{\log_2 4}{\log_2 (x^2)} + 3 \cdot \frac{\log_2 4}{\log_2 (2x)} = 3 \]

Since \( \log_2 4 = 2 \), we can substitute this into the equation:
\[ 2 \cdot \frac{2}{\log_2 (x^2)} + 3 \cdot \frac{2}{\log_2 (2x)} = 3 \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{4}{\log_2 (x^2)} + \frac{6}{\log_2 (2x)} = 3 \]

### Step 5: Express logarithms in simpler forms
We know that:
- \( \log_2 (x^2) = 2 \log_2 x \)
- \( \log_2 (2x) = \log_2 2 + \log_2 x = 1 + \log_2 x \)

Substituting these into the equation gives:
\[ \frac{4}{2 \log_2 x} + \frac{6}{1 + \log_2 x} = 3 \]

### Step 6: Let \( t = \log_2 x \)
Substituting \( t \) into the equation, we have:
\[ \frac{2}{t} + \frac{6}{1 + t} = 3 \]

### Step 7: Clear the fractions
Multiply through by \( t(1 + t) \) to eliminate the denominators:
\[ 2(1 + t) + 6t = 3t(1 + t) \]
This expands to:
\[ 2 + 2t + 6t = 3t + 3t^2 \]
Combining like terms gives:
\[ 2 + 8t = 3t + 3t^2 \]
Rearranging leads to:
\[ 3t^2 - 5t - 2 = 0 \]

### Step 8: Factor the quadratic equation
We can factor the quadratic:
\[ (3t + 1)(t - 2) = 0 \]
Setting each factor to zero gives:
1. \( 3t + 1 = 0 \) → \( t = -\frac{1}{3} \)
2. \( t - 2 = 0 \) → \( t = 2 \)

### Step 9: Solve for \( x \)
Recalling that \( t = \log_2 x \):
1. If \( t = 2 \), then \( x = 2^2 = 4 \).
2. If \( t = -\frac{1}{3} \), then \( x = 2^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{2}} \).

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