What is the value of log, base, 3, 9log
3
​
9?

The expression is given as:

\[
\log_3 \left( 9 \log_3 9 \right)
\]

### Step 1: Simplify \( \log_3 9 \)
We know that:

\[
\log_3 9 = \log_3 (3^2) = 2
\]

### Step 2: Substitute the value back into the original expression
Now, substitute \( \log_3 9 = 2 \) into the original expression:

\[
\log_3 \left( 9 \cdot 2 \right) = \log_3 (18)
\]

### Step 3: Leave the answer in logarithmic form
Thus, the simplified expression is:

\[
\log_3 18
\]

This is the final value in logarithmic form, as it cannot be simplified further without using a calculator for an approximate value.

Would you like more details on how to calculate logarithms, or do you have any questions?

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Here are 5 related questions:
1. What is \( \log_2 16 \)?
2. How do you solve \( \log_b (x) = y \) for \( x \)?
3. What is the change of base formula for logarithms?
4. Can you solve \( \log_3 27 \)?
5. How do you calculate \( \log_5 125 \)?

**Tip**: When dealing with logarithms of exponents, always try to express the number as a power of the base to simplify calculations.

Learn how to simplify and solve logarithmic expressions like log base 3 of 9. This detailed step-by-step solution walks through the properties of logarithms, exponentiation, and key formulas. Suitable for students in Grades 9-12.

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