https://file.aiquickdraw.com/mathbot/file/172734879842062qlh1cf.jpg

Let’s go through the equations in the image step by step.

### 3a. 
Solve the equation:

\[
^9 \log 25 \cdot ^5 \log 2 - ^3 \log 54
\]

Using the change of base formula:

\[
^b \log a = \frac{\log a}{\log b}
\]

1. \( ^9 \log 25 = \frac{\log 25}{\log 9} \)
2. \( ^5 \log 2 = \frac{\log 2}{\log 5} \)
3. \( ^3 \log 54 = \frac{\log 54}{\log 3} \)

Thus, the expression becomes:

\[
\frac{\log 25}{\log 9} \cdot \frac{\log 2}{\log 5} - \frac{\log 54}{\log 3}
\]

This can be calculated further numerically.

### 3b. 
Given:

- \( ^2 \log 3 = a \)
- \( ^2 \log 5 = b \)

You are asked to find \( ^2 \log \sqrt{45} \).

Since \( \sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5} \), we can break this into:

\[
^2 \log \sqrt{45} = ^2 \log (3 \sqrt{5}) = ^2 \log 3 + ^2 \log \sqrt{5}
\]

Using the properties of logarithms:

- \( ^2 \log 3 = a \)
- \( ^2 \log \sqrt{5} = \frac{1}{2} \cdot ^2 \log 5 = \frac{b}{2} \)

So,

\[
^2 \log \sqrt{45} = a + \frac{b}{2}
\]

Thus, the value of \( ^2 \log \sqrt{45} \) is \( a + \frac{b}{2} \).

---

Would you like further details on the calculations, or would you like me to evaluate specific parts? 

Here are 5 related questions to expand on this concept:

1. How does the change of base formula work for logarithms?
2. Can the properties of logarithms be applied to exponential growth problems?
3. What is the importance of logarithmic identities in simplifying expressions?
4. How can you simplify expressions with multiple logarithmic bases?
5. How can you approximate logarithmic values without a calculator?

**Tip:** When dealing with logarithms, always remember to apply properties like the product, quotient, and power rules to simplify expressions efficiently!

Selesaikan persamaan berikut: 9log25 ⋅ 5log2 − 3log54 dan Jika 2log3 = a dan 2log5 = b maka nilai 2log√45.

Learn how to simplify the logarithmic expressions 9log25 ⋅ 5log2 − 3log54 using the change of base formula. Also, find the value of 2log√45 when 2log3 = a and 2log5 = b. This solution covers key logarithmic identities and properties for advanced students.

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