To solve the equation \( \log_2 x \cdot \log_2 x - \log x - \log 5 = 1 \), let's simplify the equation step by step.

1. **Rewrite the logarithms:**  
   Recall that \( \log x = \frac{\log_2 x}{\log_2 10} \). Thus, we can express \( \log x \) in terms of \( \log_2 x \).

2. **Substituting:**  
   Let \( y = \log_2 x \). Then the equation becomes:
   \[
   y^2 - \frac{y}{\log_2 10} - \log 5 = 1
   \]

3. **Rearranging:**  
   Rearranging gives:
   \[
   y^2 - \frac{y}{\log_2 10} - \log 5 - 1 = 0
   \]

4. **Solving the quadratic equation:**  
   This is a standard quadratic equation in \( y \):
   \[
   y^2 - \frac{y}{\log_2 10} - (\log 5 + 1) = 0
   \]

   Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -\frac{1}{\log_2 10}, c = -(\log 5 + 1) \):

   \[
   b^2 = \left(-\frac{1}{\log_2 10}\right)^2 = \frac{1}{(\log_2 10)^2}
   \]
   \[
   4ac = 4 \cdot 1 \cdot -(\log 5 + 1) = -4(\log 5 + 1)
   \]

   So we calculate:
   \[
   y = \frac{\frac{1}{\log_2 10} \pm \sqrt{\frac{1}{(\log_2 10)^2} + 4(\log 5 + 1)}}{2}
   \]

5. **Finding \( x \):**  
   Once you solve for \( y \), convert back to \( x \) using \( x = 2^y \).

Would you like to proceed with the calculations, or do you have any specific questions about the steps? Here are some follow-up questions:

1. Would you like a detailed explanation of the logarithm properties used?
2. Do you want to see a numerical approximation of the solution?
3. Would you like to know how to graph the function to visualize the solution?
4. Are you interested in related logarithmic equations?
5. Do you want to explore applications of logarithmic functions in real life?

**Tip:** Always remember that changing the base of a logarithm can simplify your calculations!

Discover how to solve the logarithmic equation log2xlog2x-logx-log5=1 step by step. This guide covers logarithm properties, quadratic equations, and provides a clear method to find the solutions. Ideal for students in Grades 10-12.

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