given that log3 = x and log5 = y, express log4.5 in terms of x and y

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:

\[
\log 4.5 = 2x + y - 1
\]

---

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.

Given that log3 = x and log5 = y, express log4.5 in terms of x and y.

Learn how to express log4.5 in terms of x and y, given log3 = x and log5 = y. This problem covers important logarithmic properties such as the quotient and power rules. Ideal for high school students in Grades 10-12.

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