The expression in the image is:

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

To solve this, we'll start by simplifying the logarithms and then the cube roots.

Step 1: Simplify the logarithmic expressions

1. $\log \sqrt{8}$: $\log \sqrt{8} = \log 8^{1/2} = \frac{1}{2} \log 8$ Since $8 = 2^3$, we have: $\log 8 = \log 2^3 = 3 \log 2$ Thus, $\log \sqrt{8} = \frac{1}{2} \times 3 \log 2 = \frac{3}{2} \log 2$ The cube root is: [ \sqrt[3]{\log \sqrt{8}} = \sqrt[3]{\frac{3}{2} \log 2} ]

2. $\log 5$: The cube root is simply: $\sqrt[3]{\log 5}$

3. $\log 75$: $75$ can be factored as $75 = 3 \times 5^2$. Therefore: $\log 75 = \log (3 \times 5^2) = \log 3 + 2 \log 5$ The cube root is: [ \sqrt[3]{\log 75} = \sqrt[3]{\log 3 + 2 \log 5} ]

4. $\log 18$: $18$ can be factored as $18 = 2 \times 3^2$. Therefore: $\log 18 = \log (2 \times 3^2) = \log 2 + 2 \log 3$ The cube root is: [ \sqrt[3]{\log 18} = \sqrt[3]{\log 2 + 2 \log 3} ]

5. $\log 8$: $\log 8 = 3 \log 2$ The cube root is: [ \sqrt[3]{\log 8} = \sqrt[3]{3 \log 2} ]

Step 2: Substitute the simplified logarithmic expressions into the original expression and combine like terms

Now, substituting these back into the original expression:

[ \sqrt[3]{\frac{3}{2} \log 2} + 2 \cdot \sqrt[3]{\log 5} - \sqrt[3]{\log 3 + 2 \log 5} + \sqrt[3]{\log 2 + 2 \log 3} - \sqrt[3]{3 \log 2} ]

Step 3: Simplify further (if possible)

This expression is complex and typically doesn't simplify nicely without additional numerical values or approximations. However, without further specifics, this expression could potentially be reduced by numerical approximation methods.

Would you like to go further with this expression, perhaps using numerical approximation, or do you have any other specific questions?

Follow-up Questions:

1. Would you like to see the numerical approximation for the given expression?
2. Are you interested in a deeper exploration of logarithmic identities used here?
3. Would you like to simplify other similar logarithmic expressions?
4. Should we explore how these simplifications would work in a different logarithmic base?
5. Are you curious about the cube root functions and how they affect logarithmic expressions?

Tip:

When simplifying logarithmic expressions, always consider breaking down composite numbers into their prime factors first to make use of logarithmic identities effectively.