We can use the laws of logarithms to simplify and evaluate the expression $\log_3(9^{300})$.

Step-by-Step Solution:

1. Rewrite the logarithmic expression:

   $\log_3(9^{300})$

2. Use the power rule of logarithms, which states that $\log_b(a^n) = n \cdot \log_b(a)$. Apply this rule to move the exponent 300 in front:

   $300 \cdot \log_3(9)$

3. Simplify $\log_3(9)$:

   Since $9 = 3^2$, we can rewrite $\log_3(9)$ as $\log_3(3^2)$. Using the power rule again:

   $\log_3(3^2) = 2 \cdot \log_3(3)$

4. Evaluate $\log_3(3)$:

   By definition, $\log_3(3) = 1$, since 3 raised to the power of 1 is 3.

   $\log_3(3^2) = 2 \cdot 1 = 2$

5. Substitute back:

   Now we substitute $\log_3(9) = 2$ into the original expression:

   $300 \cdot 2 = 600$

Final Answer:

$\log_3(9^{300}) = 600$

Would you like more details, or do you have any questions?

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Here are 5 related questions:

1. How would you evaluate $\log_3(27^{100})$?
2. Can you simplify $\log_5(25^{50})$?
3. What is $\log_2(16^{20})$ using logarithmic laws?
4. How can you use the change of base formula to evaluate $\log_3(81)$?
5. Can you express $\log_3(x^5)$ in simpler form?

Tip: The power rule of logarithms is helpful when working with expressions that involve exponents, as it allows you to bring the exponent outside the logarithm.