Expand the logarithm. Assume all expressions exist and are well-defined.
Write your answer as a sum or difference of base-z logarithms or multiples of base-z logarithms. The inside of each logarithm must be a distinct constant or variable.
logz
e

3

To expand the logarithm \( \log_z\left(e^3\right) \), we can use the logarithmic power rule, which states:

\[
\log_b(x^n) = n \cdot \log_b(x)
\]

### Applying this rule:

\[
\log_z\left(e^3\right) = 3 \cdot \log_z(e)
\]

Thus, the expanded form is:

\[
3 \cdot \log_z(e)
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions for you to explore:
1. How do you expand \( \log_z(xy) \)?
2. What is the expanded form of \( \log_z\left(\frac{a}{b}\right) \)?
3. Can you simplify \( \log_z(1) \)?
4. What is the value of \( \log_z(e) \) if \( z = e \)?
5. How would you expand \( \log_z(x^5 \cdot y^2) \)?

**Tip:** Always remember that logarithms turn products into sums and powers into multiples.

Expand the logarithm. Assume all expressions exist and are well-defined. Write your answer as a sum or difference of base-z logarithms or multiples of base-z logarithms. The inside of each logarithm must be a distinct constant or variable. logz(e^3)

Explore the expansion of the logarithmic expression logz(e^3) using the power rule of logarithms. This problem demonstrates key logarithmic properties and exponentiation rules, suitable for high school students in Grades 9-12.

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