If `log_(3)27.log_x7=log_(27)x.log_(7)3`, the least value of x is

Options

A. `7^-3`
B. `3^-7`
C. `7^3`
D. `3^7`

Answer

To solve the equation given in the problem, we start with the equation:

\[ \log_{3}27 \cdot \log_{x}7 = \log_{27}x \cdot \log_{7}3 \]

### Step 1: Simplify \(\log_{3}27\) and \(\log_{27}x\)

We know that \(27 = 3^3\), so we can use the change of base formula:

\[ \log_{3}27 = \log_{3}(3^3) = 3 \]

Now, we rewrite the equation:

\[ 3 \cdot \log_{x}7 = \log_{27}x \cdot \log_{7}3 \]

### Step 2: Express \(\log_{27}x\)

Using the change of base formula again, we can express \(\log_{27}x\):

\[ \log_{27}x = \frac{\log_{3}x}{\log_{3}27} = \frac{\log_{3}x}{3} \]

Substituting this back into the equation gives:

\[ 3 \cdot \log_{x}7 = \frac{\log_{3}x}{3} \cdot \log_{7}3 \]

### Step 3: Rearranging the equation

Now we can rearrange the equation:

\[ 3 \cdot \log_{x}7 = \frac{\log_{3}x \cdot \log_{7}3}{3} \]

Multiplying both sides by 3 to eliminate the fraction:

\[ 9 \cdot \log_{x}7 = \log_{3}x \cdot \log_{7}3 \]

### Step 4: Express \(\log_{x}7\)

Using the change of base formula again, we express \(\log_{x}7\):

\[ \log_{x}7 = \frac{\log_{3}7}{\log_{3}x} \]

Substituting this into the equation gives:

\[ 9 \cdot \frac{\log_{3}7}{\log_{3}x} = \log_{3}x \cdot \log_{7}3 \]

### Step 5: Cross-multiply

Cross-multiplying gives:

\[ 9 \cdot \log_{3}7 = \log_{3}x \cdot \log_{3}x \cdot \log_{7}3 \]

This simplifies to:

\[ 9 \cdot \log_{3}7 = (\log_{3}x)^2 \cdot \log_{7}3 \]

### Step 6: Solve for \(\log_{3}x\)

Let \(y = \log_{3}x\). The equation becomes:

\[ 9 \cdot \log_{3}7 = y^2 \cdot \log_{7}3 \]

Rearranging gives:

\[ y^2 = \frac{9 \cdot \log_{3}7}{\log_{7}3} \]

### Step 7: Find the least value of \(x\)

Taking the square root of both sides:

\[ y = \sqrt{\frac{9 \cdot \log_{3}7}{\log_{7}3}} = 3 \sqrt{\frac{\log_{3}7}{\log_{7}3}} \]

Now, since \(y = \log_{3}x\), we have:

\[ \log_{3}x = 3 \sqrt{\frac{\log_{3}7}{\log_{7}3}} \]

Converting back to \(x\):

\[ x = 3^{3 \sqrt{\frac{\log_{3}7}{\log_{7}3}}} \]

Thus, the least value of \(x\) is:

\[ x = 3^{3 \sqrt{\frac{\log_{3}7}{\log_{7}3}}} \]