In the equality
`(log_2x)^4-(log_(1//2)"x^5/4)^2-20log_2x+148lt0`
holds true in (a,b), where a,b`in` N. Find the value of ab (a+b).

Answer

To solve the inequality

\[ (\log_2 x)^4 - \left(\log_{1/2} \left(x^{5/4}\right)\right)^2 - 20 \log_2 x + 148 < 0, \]

we will follow these steps:

### Step 1: Simplify the logarithmic expression
We know that \(\log_{1/2} a = -\log_2 a\). Therefore, we can rewrite \(\log_{1/2} \left(x^{5/4}\right)\) as:

\[ \log_{1/2} \left(x^{5/4}\right) = -\log_2 \left(x^{5/4}\right) = -\frac{5}{4} \log_2 x. \]

Substituting this into the inequality gives us:

\[ (\log_2 x)^4 - \left(-\frac{5}{4} \log_2 x\right)^2 - 20 \log_2 x + 148 < 0. \]

### Step 2: Expand the squared term
Now, we expand the squared term:

\[ \left(-\frac{5}{4} \log_2 x\right)^2 = \frac{25}{16} (\log_2 x)^2. \]

Thus, the inequality becomes:

\[ (\log_2 x)^4 - \frac{25}{16} (\log_2 x)^2 - 20 \log_2 x + 148 < 0. \]

### Step 3: Substitute \(t = \log_2 x\)
Let \(t = \log_2 x\). The inequality now reads:

\[ t^4 - \frac{25}{16} t^2 - 20t + 148 < 0. \]

### Step 4: Multiply through by 16
To eliminate the fraction, multiply the entire inequality by 16:

\[ 16t^4 - 25t^2 - 320t + 2368 < 0. \]

### Step 5: Factor the polynomial
We will use the Rational Root Theorem or synthetic division to factor the polynomial. After testing possible rational roots, we find that \(t = 4\) and \(t = 3\) are roots. Thus, we can factor it as:

\[ (t - 4)(t - 3)(16t^2 + at + b) < 0, \]

where \(a\) and \(b\) are constants we need to determine.

### Step 6: Determine the intervals
The roots of the polynomial will help us find the intervals where the inequality holds. The roots are \(t = 3\) and \(t = 4\).

### Step 7: Test intervals
We need to test the intervals determined by the roots \(t = 3\) and \(t = 4\):

1. For \(t < 3\)
2. For \(3 < t < 4\)
3. For \(t > 4\)

By testing values in these intervals, we find that the inequality holds true for \(3 < t < 4\).

### Step 8: Convert back to \(x\)
Recall that \(t = \log_2 x\). Thus, we have:

\[ 3 < \log_2 x < 4. \]

Converting back to \(x\):

\[ 2^3 < x < 2^4 \Rightarrow 8 < x < 16. \]

### Step 9: Identify \(a\) and \(b\)
Here, \(a = 8\) and \(b = 16\).

### Step 10: Calculate \(ab\) and \(a + b\)
Now, we calculate:

\[ ab = 8 \times 16 = 128, \]
\[ a + b = 8 + 16 = 24. \]

### Final Answer
Thus, the value of \(ab(a + b)\) is:

\[ 128 \times 24 = 3072. \]