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The given expression is:

\[
\ln x + 4 \ln y - 5 \ln z
\]

We need to rewrite this as a single logarithm of the form \(\ln A\).

### Step-by-step solution:

1. Use the logarithmic property \( a \ln b = \ln(b^a) \) to handle the coefficients in front of the logarithms.
   - \( 4 \ln y = \ln y^4 \)
   - \( 5 \ln z = \ln z^5 \)

Thus, the expression becomes:

\[
\ln x + \ln y^4 - \ln z^5
\]

2. Use the property of logarithms that states: \(\ln a + \ln b = \ln(ab)\) and \(\ln a - \ln b = \ln(a/b)\).
   - Combine the logarithms: 

\[
\ln \left( \frac{x y^4}{z^5} \right)
\]

Thus, the expression is now written as a single logarithm, \(\ln A\), where:

\[
A = \frac{x y^4}{z^5}
\]

So, the final answer is:

\[
A = \frac{x y^4}{z^5}
\]

Would you like further details or have any questions? Here are some related questions to consider:

1. What other logarithmic properties could be useful for simplifying expressions?
2. How does the power rule for logarithms apply in different contexts?
3. What happens when logarithms have different bases?
4. Can you explain why logarithms are useful for solving exponential equations?
5. How would you solve an equation where the logarithms on both sides are equal?

**Tip**: Always remember the key logarithmic identities, such as the product rule, quotient rule, and power rule, to simplify expressions effectively.

Rewrite the expression ln(x) + 4ln(y) - 5ln(z) as a single logarithm, ln(A). What is A?

Learn how to simplify ln(x) + 4ln(y) - 5ln(z) as a single logarithmic expression ln(A) using logarithmic properties like the power, product, and quotient rules. This problem covers key algebraic concepts, suitable for students in Grades 9-12.

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