The expression ln3 + In9 + In 27 in simplified form is a ln b.

To simplify the expression \( \ln 3 + \ln 9 + \ln 27 \), we can use the **logarithmic addition rule**:

\[
\ln a + \ln b + \ln c = \ln (a \cdot b \cdot c)
\]

### Step-by-step simplification:
1. Using the rule for adding logarithms:
   \[
   \ln 3 + \ln 9 + \ln 27 = \ln (3 \cdot 9 \cdot 27)
   \]
2. Multiply the numbers inside the logarithm:
   \[
   3 \cdot 9 = 27, \quad 27 \cdot 27 = 729
   \]
   So, we get:
   \[
   \ln (729)
   \]
3. Finally, express 729 as a power of 3:
   \[
   729 = 3^6
   \]
4. Apply the logarithmic power rule \( \ln (a^b) = b \ln a \):
   \[
   \ln (3^6) = 6 \ln 3
   \]

Thus, the simplified form of \( \ln 3 + \ln 9 + \ln 27 \) is \( 6 \ln 3 \).

### Final Answer:
\[
a = 6, \quad b = 3
\]
So, the simplified form is \( 6 \ln 3 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you simplify the sum of multiple logarithms?
2. What are the basic properties of logarithms?
3. How can you express logarithms in terms of powers of a number?
4. What is the product rule of logarithms?
5. How can you convert logarithmic expressions into exponential form?

**Tip**: When adding logarithms, always check if you can simplify the argument inside the logarithms first!

The expression ln3 + ln9 + ln27 in simplified form is a ln b.

Learn how to simplify the logarithmic expression ln 3 + ln 9 + ln 27 using the logarithmic addition and power rules. The problem explores key logarithmic concepts, suitable for students in Grades 9-12.

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