To expand the logarithmic expression using properties of logarithms, we need to apply the following logarithmic rules:

1. Logarithm of a quotient: $\log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N)$
2. Logarithm of a product: $\log_b(MN) = \log_b(M) + \log_b(N)$
3. Logarithm of a power: $\log_b(M^n) = n \cdot \log_b(M)$
4. Logarithm of a root: $\log_b(\sqrt[n]{M}) = \frac{1}{n} \cdot \log_b(M)$

Now, let's apply these rules to the given expression:

$\log_d \left( \frac{\sqrt{a} b^5}{c^3} \right)$

Step-by-step breakdown:

1. Apply the quotient rule: $\log_d \left( \frac{\sqrt{a} b^5}{c^3} \right) = \log_d(\sqrt{a} b^5) - \log_d(c^3)$

2. Apply the product rule to $\log_d(\sqrt{a} b^5)$: $\log_d(\sqrt{a} b^5) = \log_d(\sqrt{a}) + \log_d(b^5)$

3. Apply the power rule to $\log_d(b^5)$: $\log_d(b^5) = 5 \cdot \log_d(b)$

4. Apply the power rule to $\log_d(\sqrt{a})$. Since $\sqrt{a} = a^{1/2}$, we get: $\log_d(\sqrt{a}) = \frac{1}{2} \cdot \log_d(a)$

5. Apply the power rule to $\log_d(c^3)$: $\log_d(c^3) = 3 \cdot \log_d(c)$

Putting it all together:

$\log_d \left( \frac{\sqrt{a} b^5}{c^3} \right) = \frac{1}{2} \cdot \log_d(a) + 5 \cdot \log_d(b) - 3 \cdot \log_d(c)$

So, the expanded form of the expression is: $\frac{1}{2} \log_d(a) + 5 \log_d(b) - 3 \log_d(c)$

This is the fully expanded logarithmic expression.

---

Would you like any more details on this process or have any other questions? Here are some related questions you might find interesting:

1. How can you apply logarithmic properties to solve logarithmic equations?
2. What happens when you apply the logarithmic properties to solve for one of the variables inside a logarithm?
3. How do natural logarithms (ln) differ from common logarithms (log base 10)?
4. Can you simplify logarithmic expressions with different bases?
5. How do logarithmic properties assist in integrating functions involving logarithms?

Tip: When expanding logarithmic expressions, always break down the components step by step, using the appropriate properties for each part (product, quotient, and power).