The sum of all positive numbers x such that `(log_(x)3)(log_(x) 9)+2=5 log_(x) 3` is a value between

Options

A. 5 and 7
B. 7 and 9
C. 9 and 11
D. 11 and 13

Answer

To solve the problem, we need to find the sum of all positive numbers \( x \) such that:

\[ (\log_{x} 3)(\log_{x} 9) + 2 = 5 \log_{x} 3 \]

### Step 1: Rewrite the logarithmic expressions

We know that \( \log_{x} 9 = \log_{x} (3^2) = 2 \log_{x} 3 \). Let's denote \( t = \log_{x} 3 \). Then, we can rewrite the equation as:

\[ t(2t) + 2 = 5t \]

### Step 2: Simplify the equation

Substituting \( \log_{x} 9 \) into the equation gives us:

\[ 2t^2 + 2 = 5t \]

### Step 3: Rearrange the equation

Rearranging the equation leads to:

\[ 2t^2 - 5t + 2 = 0 \]

### Step 4: Factor the quadratic equation

Now, we can factor the quadratic equation. We look for two numbers that multiply to \( 2 \times 2 = 4 \) and add to \( -5 \). The numbers are \( -4 \) and \( -1 \):

\[ 2t^2 - 4t - t + 2 = 0 \]

Factoring gives us:

\[ (2t - 1)(t - 2) = 0 \]

### Step 5: Solve for \( t \)

Setting each factor to zero gives us:

1. \( 2t - 1 = 0 \) → \( t = \frac{1}{2} \)
2. \( t - 2 = 0 \) → \( t = 2 \)

### Step 6: Find \( x \) from \( t \)

Recall that \( t = \log_{x} 3 \). We can convert this back to \( x \):

1. For \( t = \frac{1}{2} \):
\[ \log_{x} 3 = \frac{1}{2} \implies x^{\frac{1}{2}} = 3 \implies x = 3^2 = 9 \]

2. For \( t = 2 \):
\[ \log_{x} 3 = 2 \implies x^2 = 3 \implies x = 3^{\frac{1}{2}} = \sqrt{3} \]

### Step 7: Sum the values of \( x \)

Now we have two positive values for \( x \):
- \( x_1 = 9 \)
- \( x_2 = \sqrt{3} \)

The sum of these values is:

\[ 9 + \sqrt{3} \]

### Step 8: Approximate the sum

To approximate \( \sqrt{3} \):
\[ \sqrt{3} \approx 1.732 \]
Thus,
\[ 9 + \sqrt{3} \approx 9 + 1.732 = 10.732 \]

### Conclusion

The sum of all positive numbers \( x \) is approximately \( 10.732 \), which lies between \( 9 \) and \( 11 \).