`log_(16)x^(3)+(log_(2)sqrt(x))^(2) lt 1` iff x lies in

Options

A. (2,16)
B. `(0,1//16)`
C. `(1//16,2)`
D. (0,2)

Answer

To solve the inequality \( \log_{16}(x^3) + (\log_{2}(\sqrt{x}))^2 < 1 \), we can follow these steps:

### Step 1: Rewrite the logarithmic expressions
We can use the change of base formula for logarithms. Since \( 16 = 2^4 \), we can rewrite \( \log_{16}(x^3) \) as:
\[ \log_{16}(x^3) = \frac{\log_{2}(x^3)}{\log_{2}(16)} = \frac{\log_{2}(x^3)}{4} = \frac{3}{4} \log_{2}(x) \]
And for \( \log_{2}(\sqrt{x}) \):
\[ \log_{2}(\sqrt{x}) = \log_{2}(x^{1/2}) = \frac{1}{2} \log_{2}(x) \]
Thus, \( (\log_{2}(\sqrt{x}))^2 = \left(\frac{1}{2} \log_{2}(x)\right)^2 = \frac{1}{4} (\log_{2}(x))^2 \).

### Step 2: Substitute back into the inequality
Substituting these expressions back into the inequality gives:
\[ \frac{3}{4} \log_{2}(x) + \frac{1}{4} (\log_{2}(x))^2 < 1 \]

### Step 3: Multiply through by 4 to eliminate the fraction
To eliminate the fraction, multiply the entire inequality by 4:
\[ 3 \log_{2}(x) + (\log_{2}(x))^2 < 4 \]

### Step 4: Rearrange the inequality
Rearranging the inequality gives:
\[ (\log_{2}(x))^2 + 3 \log_{2}(x) - 4 < 0 \]

### Step 5: Let \( t = \log_{2}(x) \)
Letting \( t = \log_{2}(x) \), we rewrite the inequality as:
\[ t^2 + 3t - 4 < 0 \]

### Step 6: Factor the quadratic expression
We can factor the quadratic:
\[ (t + 4)(t - 1) < 0 \]

### Step 7: Find the critical points
The critical points are \( t = -4 \) and \( t = 1 \).

### Step 8: Determine the intervals
We analyze the sign of the product \( (t + 4)(t - 1) \):
- For \( t < -4 \): both factors are negative, product is positive.
- For \( -4 < t < 1 \): one factor is positive and the other is negative, product is negative.
- For \( t > 1 \): both factors are positive, product is positive.

Thus, the solution to the inequality is:
\[ -4 < t < 1 \]

### Step 9: Substitute back for \( x \)
Recall that \( t = \log_{2}(x) \):
\[ -4 < \log_{2}(x) < 1 \]

### Step 10: Convert back to exponential form
This corresponds to:
\[ 2^{-4} < x < 2^{1} \]
Calculating these values gives:
\[ \frac{1}{16} < x < 2 \]

### Final Answer
Thus, the solution for \( x \) is:
\[ \frac{1}{16} < x < 2 \]