https://file.aiquickdraw.com/mathbot/file/1725523193663538hyl0d.jpg

The equation shown is:

\[
\log_9 9^{\log_3 3^3} = x
\]

### Step-by-step solution:

1. **Simplify the exponent \(\log_3 3^3\):**

   \[
   \log_3 3^3 = 3
   \]
   This is because the logarithmic property \(\log_b b^x = x\) tells us that the exponent "pops out" of the logarithm when the base is the same.

   So, the equation becomes:

   \[
   \log_9 9^3 = x
   \]

2. **Simplify \(\log_9 9^3\):**

   Using the same logarithmic property \(\log_b b^x = x\):

   \[
   \log_9 9^3 = 3
   \]

Thus, the equation simplifies to:

\[
x = 3
\]

### Final Answer:
\[
x = 3
\]

Do you have any questions or would you like more details?

### Related Questions:
1. How do you convert between logarithms of different bases?
2. What is the relationship between logarithms and exponents?
3. How can you simplify nested logarithms in an expression?
4. What are the key properties of logarithms that are frequently used in solving equations?
5. How can you apply the change of base formula to evaluate complex logarithmic expressions?

### Tip:
When simplifying logarithmic expressions, always look for opportunities to apply properties like \(\log_b b^x = x\) and the change of base formula.

Learn how to simplify the logarithmic expression log_9 9^(log_3 3^3) with a step-by-step solution. This problem covers logarithmic and exponential properties, including the power rule of logarithms and the exponentiation property. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation