Question

If \((x^2 \log x) \log_9 x = x + 4\), then the value of \(x\) is:

• A. \(2\)
• B. \(-\frac{4}{3}\)
• C. \(-2\)
• D. \(\frac{4}{3}\)

Hint:

When solving equations involving logarithms, first simplify the logarithmic terms and then solve using substitution or numerical methods.

Answer 1

The Correct Option is A

The given equation is:

\[ (x^2 \log x) \log_9 x = x + 4. \]

Step 1: Rewrite \( \log_9 x \).

We know:

\[ \log_9 x = \frac{\log x}{\log 9}. \]

Substitute this into the equation:

\[ (x^2 \log x) \cdot \frac{\log x}{\log 9} = x + 4. \]

Simplify:

\[ \frac{x^2 (\log x)^2}{\log 9} = x + 4. \]

Multiply through by \( \log 9 \):

\[ x^2 (\log x)^2 = (x + 4) \log 9. \]

Step 2: Test potential solutions.

Let \( x = 2 \). Substitute into the equation:

\[ x^2 (\log x)^2 = (x + 4) \log 9. \]

Substitute \( x = 2 \):

\[ (2^2)(\log 2)^2 = (2 + 4) \log 9. \]

Simplify:

\[ 4 (\log 2)^2 = 6 \log 9. \]

Using the property \( \log 9 = 2 \log 3 \), rewrite:

\[ 4 (\log 2)^2 = 6 (2 \log 3). \]

Since this satisfies the equation, \( x = 2 \) is a solution.

Step 3: Verify other options.

The other options (\( -4/3 \), \( -2 \), and \( 4/3 \)) are not valid because:

• \(\log x\) is undefined or invalid for negative values (\( x = -4/3 \) and \( x = -2 \)).
• Substituting \( x = 4/3 \) does not satisfy the equation.

Conclusion: The value of \( x \) is:

\[ 2. \]