If `(log_(3)x)(log_(x)2x)(log_(2x)y)=log_(x)x^(2)`, then what is y equal to?

Options

A. 4.5
B. 9
C. 18
D. 27

Explanation

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

### Step 1: Rewrite the equation using change of base formula
Using the change of base formula, we can express the logarithms in terms of natural logarithms (or common logarithms). The change of base formula states that \(\log_{a}b = \frac{\log b}{\log a}\).

Thus, we rewrite each term:
\[ \log_{3}x = \frac{\log x}{\log 3}, \quad \log_{x}(2x) = \frac{\log(2x)}{\log x}, \quad \log_{2x}y = \frac{\log y}{\log(2x)} \]
Substituting these into the equation gives:
\[ \left(\frac{\log x}{\log 3}\right) \left(\frac{\log(2x)}{\log x}\right) \left(\frac{\log y}{\log(2x)}\right) = \log_{x}x^{2} \]

### Step 2: Simplify the left-hand side
Notice that \(\log(2x) = \log 2 + \log x\). Therefore, we can rewrite the left-hand side:
\[ \left(\frac{\log x}{\log 3}\right) \left(\frac{\log 2 + \log x}{\log x}\right) \left(\frac{\log y}{\log(2x)}\right) \]
This simplifies to:
\[ \frac{\log y \cdot (\log 2 + \log x)}{\log 3 \cdot \log(2x)} \]

### Step 3: Simplify the right-hand side
The right-hand side, \(\log_{x}x^{2}\), can be simplified using the property of logarithms:
\[ \log_{x}x^{2} = 2 \]

### Step 4: Set the equation
Now we have:
\[ \frac{\log y \cdot (\log 2 + \log x)}{\log 3 \cdot \log(2x)} = 2 \]

### Step 5: Solve for \(\log y\)
We can rearrange this equation:
\[ \log y \cdot (\log 2 + \log x) = 2 \cdot \log 3 \cdot \log(2x) \]
Now, substituting \(\log(2x) = \log 2 + \log x\):
\[ \log y \cdot (\log 2 + \log x) = 2 \cdot \log 3 \cdot (\log 2 + \log x) \]

### Step 6: Divide both sides by \((\log 2 + \log x)\)
Assuming \((\log 2 + \log x) \neq 0\), we can divide both sides:
\[ \log y = 2 \cdot \log 3 \]

### Step 7: Exponentiate to solve for \(y\)
To find \(y\), we exponentiate both sides:
\[ y = 3^{2} = 9 \]

Thus, the value of \(y\) is \(9\).