We are given:

$\log 3 = x \quad \text{and} \quad \log 5 = y$

We need to express $\log 4.5$ in terms of $x$ and $y$.

Start by expressing $4.5$ as a product:

$4.5 = \frac{9}{2}$

Thus,

$\log 4.5 = \log \left( \frac{9}{2} \right)$

Using the logarithmic property:

$\log \left( \frac{a}{b} \right) = \log a - \log b$

We can write:

$\log 4.5 = \log 9 - \log 2$

Now, $\log 9$ can be written as $\log 9 = \log (3^2)$, and using the logarithmic power rule $\log (a^b) = b \log a$, we get:

$\log 9 = 2 \log 3$

Substitute $\log 3 = x$:

$\log 9 = 2x$

So,

$\log 4.5 = 2x - \log 2$

Next, we need to express $\log 2$ in terms of $x$ and $y$. Since $\log 10 = 1$, we use the approximation:

$\log 10 = \log 2 + \log 5$

Thus:

$1 = \log 2 + y$

Solving for $\log 2$:

$\log 2 = 1 - y$

Finally, substitute this into the expression for $\log 4.5$:

$\log 4.5 = 2x - (1 - y)$

Simplify:

$\log 4.5 = 2x - 1 + y$

Thus, the expression for $\log 4.5$ in terms of $x$ and $y$ is:

$\log 4.5 = 2x + y - 1$

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Would you like further explanation on any of these steps or details on logarithmic properties?

Here are 5 related questions:

1. How can you express $\log (a \cdot b)$ using logarithmic rules?
2. What is the logarithmic power rule, and how does it apply to $\log(3^2)$?
3. How would you express $\log (7.5)$ in terms of $x$ and $y$?
4. What is the relationship between common logarithms (base 10) and natural logarithms (base $e$)?
5. How does the change of base formula work in logarithms?

Tip: When simplifying logarithmic expressions, always look for opportunities to apply the product, quotient, and power rules.