\ln ((e^(7))/(3))
Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.

We are given the expression:

\[
\ln \left( \frac{e^7}{3} \right)
\]

### Step 1: Apply the logarithmic property for division
The logarithmic property for division is:

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

Using this property on the given expression:

\[
\ln \left( \frac{e^7}{3} \right) = \ln(e^7) - \ln(3)
\]

### Step 2: Apply the logarithmic property for exponents
The logarithmic property for exponents is:

\[
\ln(A^k) = k \cdot \ln(A)
\]

Applying this property to \(\ln(e^7)\):

\[
\ln(e^7) = 7 \cdot \ln(e)
\]

Since \(\ln(e) = 1\), we have:

\[
\ln(e^7) = 7
\]

### Step 3: Combine the terms
Now, substitute back into the original expression:

\[
\ln \left( \frac{e^7}{3} \right) = 7 - \ln(3)
\]

This is the expanded form of the logarithmic expression.

### Final Answer:
\[
\ln \left( \frac{e^7}{3} \right) = 7 - \ln(3)
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Expand \(\ln \left( \frac{5}{x^2} \right)\) using properties of logarithms.
2. Simplify \(\ln \left( e^{4x} \right)\).
3. Expand \(\ln \left( \frac{2x}{y^3} \right)\).
4. Evaluate \(\ln \left( e^{10} \right)\).
5. Simplify \(\ln \left( e^{2} \cdot 4 \right)\).

**Tip:** Always remember that \(\ln(e) = 1\), which makes expressions with \(e\) easier to simplify!

Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible. \ln \left( \frac{e^7}{3} \right)

Learn how to expand the logarithmic expression \ln \left( \frac{e^7}{3} \right) using properties of logarithms. This step-by-step guide covers key concepts such as the logarithmic property for division and exponents, ideal for students in Grades 10-12.

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