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We are given:

\[
\log_3 h = x \quad \text{and} \quad \log_3 k = y
\]

We are tasked with expressing \(\log_{27}\left(\frac{9k}{h}\right)\) in terms of \(x\) and \(y\).

### Step 1: Simplify the base of the logarithm

We can first rewrite the logarithm \(\log_{27}\left(\frac{9k}{h}\right)\) in terms of base 3, because \(27 = 3^3\). Using the change of base formula, we have:

\[
\log_{27}\left(\frac{9k}{h}\right) = \frac{1}{3} \log_3 \left(\frac{9k}{h}\right)
\]

### Step 2: Break down the logarithm

Next, apply the logarithmic property that states:

\[
\log_b \left(\frac{A}{B}\right) = \log_b A - \log_b B
\]

Thus:

\[
\log_3 \left(\frac{9k}{h}\right) = \log_3 (9k) - \log_3 h
\]

### Step 3: Further simplify

Now simplify \(\log_3 (9k)\) using the property:

\[
\log_3 (9k) = \log_3 9 + \log_3 k
\]

Since \(\log_3 9 = 2\) (because \(9 = 3^2\)), we get:

\[
\log_3 (9k) = 2 + \log_3 k = 2 + y
\]

### Step 4: Combine everything

We now substitute back:

\[
\log_3 \left(\frac{9k}{h}\right) = (2 + y) - x
\]

Thus:

\[
\log_{27}\left(\frac{9k}{h}\right) = \frac{1}{3} [(2 + y) - x]
\]

### Final expression:

\[
\log_{27}\left(\frac{9k}{h}\right) = \frac{1}{3} (2 + y - x)
\]

This is the logarithmic expression in terms of \(x\) and \(y\).

Would you like further clarification, or do you have any questions?

Here are 5 related questions:
1. How can you convert logarithms with different bases?
2. What are the key properties of logarithms used in simplifications?
3. How do you apply the change of base formula to other logarithmic bases?
4. How can you use the same approach to express \(\log_9 \left(\frac{h}{k}\right)\)?
5. What are some other logarithmic identities involving sums and differences?

**Tip:** Always look for opportunities to apply logarithmic identities, such as the power rule, product rule, or change of base formula, to simplify complex expressions.

Diberi bahawa log_3 h = x dan log_3 k = y, ungkapkan log_27(9k/h) dalam sebutan x dan y

Learn how to express log_27(9k/h) using the variables x and y, where log_3 h = x and log_3 k = y. This problem involves logarithmic properties, including the change of base formula and product rule, suitable for students in Grades 10-12.

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