https://file.aiquickdraw.com/mathbot/file/1726725555877z7tkqaek.jpg

The expression shown in the image is:

\[
5 \log 16 \cdot x^{2 \log \sqrt{3}}, 9 \log 5
\]

### Simplifying the components:

1. **First part: \( 5 \log 16 \cdot x^{2 \log \sqrt{3}} \)**

   - \( 5 \log 16 \) is a constant multiplication term. We know that \( \log 16 = \log (2^4) = 4 \log 2 \). Thus:
     \[
     5 \log 16 = 5 \cdot 4 \log 2 = 20 \log 2
     \]
   - Now, simplify \( x^{2 \log \sqrt{3}} \). Since \( \sqrt{3} = 3^{1/2} \), we can express \( \log \sqrt{3} = \frac{1}{2} \log 3 \). Therefore:
     \[
     x^{2 \log \sqrt{3}} = x^{2 \cdot \frac{1}{2} \log 3} = x^{\log 3}
     \]
     Using the logarithmic property \( x^{\log a} = a^{\log x} \), we can simplify:
     \[
     x^{\log 3} = 3^{\log x}
     \]

   Hence, the entire first part simplifies to:
   \[
   20 \log 2 \cdot 3^{\log x}
   \]

2. **Second part: \( 9 \log 5 \)**
   
   This is simply:
   \[
   9 \log 5
   \]

Thus, the simplified form of the expression is:

\[
20 \log 2 \cdot 3^{\log x}, 9 \log 5
\]

Would you like more details on logarithmic properties or have any questions?

### Related Questions:
1. How do you apply the power rule of logarithms in other scenarios?
2. Can you solve equations involving logarithmic exponents?
3. How would you solve for \( x \) if given \( 3^{\log x} = a \)?
4. What is the change of base formula for logarithms?
5. How do you evaluate logarithmic expressions involving different bases?

### Tip:
Remember, logarithmic properties such as \( \log (a^b) = b \log a \) are incredibly useful for simplifying complex expressions.

Solve the expression involving logarithms in the image.

This step-by-step guide simplifies a logarithmic expression involving exponents and roots. Learn how to apply logarithmic properties such as log(a^b) = b log(a) and exponent rules for logarithms. Suitable for students in Grades 10-12.

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