https://file.aiquickdraw.com/mathbot/file/17254655700966loyhwm6.jpg

The expression provided in the image is:

\[
^3\log \sqrt{8} + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 - ^3\log 8
\]

Let's break this down step by step:

### Step 1: Simplify the logarithmic terms.
Recall that:
\[
^3\log a^b = b \times ^3\log a
\]
\[
\sqrt{8} = 8^{1/2}
\]

Thus:
\[
^3\log \sqrt{8} = \frac{1}{2} \times ^3\log 8
\]

So the expression becomes:
\[
\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 - ^3\log 8
\]

### Step 2: Combine the logarithmic terms.

Now, combine the terms involving \( ^3\log 8 \):

\[
\frac{1}{2} \times ^3\log 8 - ^3\log 8 = -\frac{1}{2} \times ^3\log 8
\]

Now, we have:
\[
-\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18
\]

### Step 3: Use the properties of logarithms.

We know the logarithm of a product is the sum of the logarithms:
\[
^3\log a - ^3\log b = ^3\log \left(\frac{a}{b}\right)
\]
Thus:
\[
-\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18
\]
\[
= -\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 + ^3\log \left(\frac{18}{75}\right)
\]

### Step 4: Simplify the final expression.
\[
\frac{18}{75} = \frac{6}{25} = \frac{2 \times 3}{5^2}
\]
So:
\[
^3\log \left(\frac{18}{75}\right) = ^3\log \left(\frac{6}{25}\right) = ^3\log \left(\frac{2 \times 3}{5^2}\right)
\]

Thus, the expression simplifies further to:
\[
-\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 + ^3\log \left(\frac{2 \times 3}{25}\right)
\]

And since \( \frac{1}{25} \) can be expressed as \( 5^{-2} \):
\[
= -\frac{1}{2} \times ^3\log 8 + 2 \times ^3\log 5 + ^3\log (2 \times 3) - 2 \times ^3\log 5
\]
\[
= -\frac{1}{2} \times ^3\log 8 + ^3\log 6
\]

Given the nature of this logarithm, the expression cannot be simplified further using basic algebraic techniques.

**Final Answer:**
\[
-\frac{1}{2} \times ^3\log 8 + ^3\log 6
\]

Would you like more details on any step or do you have any further questions?

---

### Relative Questions:
1. How can you simplify logarithms with different bases?
2. What is the significance of the properties of logarithms in simplifying expressions?
3. How would the expression change if the logarithm base were different?
4. What are some common logarithm identities used in simplifications?
5. How can logarithms be applied to real-world problems, such as in finance or engineering?

### Tip:
Always check if a logarithmic expression can be further simplified by using properties such as the product, quotient, or power rules before calculating.

^3\log \sqrt{8} + 2 \times ^3\log 5 - ^3\log 75 + ^3\log 18 - ^3\log 8 =

Explore how to simplify a complex logarithmic expression with base 3 involving various arithmetic operations. The solution uses key logarithmic properties and algebraic simplification techniques. Suitable for students in Grades 10-12.

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