Let's solve the logarithmic expression given in the image:

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

Step 1: Simplify each logarithmic term.

1. $^3\log \sqrt[3]{8} = ^3\log 2$

   Explanation: $\sqrt[3]{8} = 2$ because $2^3 = 8$.

2. $2 \cdot ^3\log 5$ is already in its simplest form.

3. $^3\log 75$ can be broken down using the properties of logarithms:

   $^3\log 75 = ^3\log(3 \times 25) = ^3\log 3 + ^3\log 25 = ^3\log 3 + 2 \cdot ^3\log 5$

4. $^3\log 18$ can be simplified as:

   $^3\log 18 = ^3\log(2 \times 9) = ^3\log 2 + 2 \cdot ^3\log 3$

5. $^3\log 8$ simplifies to:

   $^3\log 8 = ^3\log 2^3 = 3 \cdot ^3\log 2$

Step 2: Substitute and combine like terms.

Substitute all the simplified terms back into the original expression:

$^3\log 2 + 2 \cdot ^3\log 5 - (^3\log 3 + 2 \cdot ^3\log 5) + (^3\log 2 + 2 \cdot ^3\log 3) - 3 \cdot ^3\log 2$

Combine like terms:

$(^3\log 2 + ^3\log 2 - 3 \cdot ^3\log 2) + (2 \cdot ^3\log 5 - 2 \cdot ^3\log 5) + (^3\log 3 + 2 \cdot ^3\log 3 - ^3\log 3)$

This simplifies to:

$(0) + (0) + (^3\log 3)$

Thus, the expression simplifies to:

$^3\log 3$

The final answer is $^3\log 3$.

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

Here are 5 related questions:

1. How would the expression change if the base of the logarithms was different?
2. Can the properties of logarithms be applied to logarithms with any base?
3. How do you simplify an expression involving multiple logarithmic terms?
4. What are some real-world applications of logarithms?
5. How do logarithmic identities help in solving complex equations?

Tip: Always double-check the simplification of logarithmic terms before combining them. This reduces the risk of errors.