The image shows a mathematical expression involving logarithms. The expression reads:

$\log \frac{3}{2} + \log 4 \sqrt{2}$

Step-by-step solution:

1. Apply the logarithmic product rule:

The logarithmic product rule states: $\log(a) + \log(b) = \log(a \cdot b)$ Apply this rule to the given expression: $\log \frac{3}{2} + \log 4\sqrt{2} = \log \left( \frac{3}{2} \cdot 4\sqrt{2} \right)$

2. Simplify the product inside the logarithm:

$\frac{3}{2} \cdot 4\sqrt{2} = \frac{3 \cdot 4\sqrt{2}}{2} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}$

Thus, the expression becomes: $\log(6\sqrt{2})$

Final Answer:

$\log(6\sqrt{2})$

Let me know if you'd like further details or assistance! 😊

Related Questions:

1. What are the properties of logarithms, and how are they applied in calculations?
2. How do you simplify products or quotients inside a logarithmic expression?
3. What is the difference between common logarithms ($\log$) and natural logarithms ($\ln$)?
4. How can you express $\log(6\sqrt{2})$ in terms of its individual components ($\log(6)$ and $\log(\sqrt{2})$)?
5. How would the solution change if the base of the logarithm was not 10?

Tip:

Always remember the logarithmic power rule: $\log(a^b) = b \log(a)$, which can simplify terms involving roots or exponents!