Expand the logarithm as much as possible. Rewrite the expression as a sum, difference, or product of logs. ln(1/25^k) Enclose arguments of functions in parentheses and include a multiplication sign between terms. For example, c*ln(h) .

We are given the expression \( \ln\left( \frac{1}{25^k} \right) \) and asked to rewrite it using logarithmic properties. Let's break it down step by step:

### Step 1: Apply the quotient rule for logarithms
The logarithmic quotient rule states:
\[
\ln\left( \frac{a}{b} \right) = \ln(a) - \ln(b)
\]
Using this rule, we can rewrite the expression:
\[
\ln\left( \frac{1}{25^k} \right) = \ln(1) - \ln(25^k)
\]

### Step 2: Simplify \( \ln(1) \)
Since \( \ln(1) = 0 \), we get:
\[
0 - \ln(25^k) = -\ln(25^k)
\]

### Step 3: Apply the power rule for logarithms
The power rule states:
\[
\ln(a^b) = b \cdot \ln(a)
\]
Using this rule, we can rewrite \( \ln(25^k) \) as:
\[
\ln(25^k) = k \cdot \ln(25)
\]

Thus, the expression becomes:
\[
-\ln(25^k) = -k \cdot \ln(25)
\]

### Final Answer:
\[
\ln\left( \frac{1}{25^k} \right) = -k \cdot \ln(25)
\]

Would you like further details on any of the steps?

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Here are five related questions to explore:

1. What is the value of \( \ln(25) \) in terms of natural logarithms of prime factors?
2. How would the expansion change if the base of the logarithm were 10 instead of \( \ln \)?
3. How do you expand \( \ln\left( \frac{x^2}{y^3} \right) \)?
4. What is the derivative of \( \ln(25^k) \) with respect to \( k \)?
5. Can you apply the same logarithmic rules to simplify \( \ln\left( \frac{(x+2)}{y^k} \right) \)?

**Tip**: Always remember the power rule in logarithms—it’s handy for simplifying powers inside a logarithmic expression!

Expand the logarithm as much as possible. Rewrite the expression as a sum, difference, or product of logs. ln(1/25^k)

Learn how to expand the logarithmic expression ln(1/25^k) using logarithmic properties like the quotient and power rules. This step-by-step explanation is suitable for students in Grades 10-12 and covers key concepts in algebra and logarithms.

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