To determine the exponent \( x \) that makes the two expressions \( (33)^x = 324 \) equivalent, we can rewrite 324 as a power of 33.
First, we can calculate the logarithm of both sides:
\[ x \cdot \log(33) = \log(324) \]
Now, we can solve for \( x \):
\[ x = \frac{\log(324)}{\log(33)} \]
Next, we can calculate these logarithms:
- Using a calculator or logarithm tables, we find:
- \( \log(324) \approx 2.51082562376 \)
- \( \log(33) \approx 1.51851393987 \)
Now, substituting these values into the equation for \( x \):
\[ x \approx \frac{2.51082562376}{1.51851393987} \approx 1.65 \]
So, the exponent \( x \) is approximately \( 1.65 \).
However, since none of the provided options (15, 8, 21, 27) are equivalent to the computed exponent, let's try expressing 324 in terms of the base 33 to check if it matches with any of the options.
The value 324 can also be seen as:
\[ 324 = 18^2 = (3^2 \cdot 2)^2 = 3^4 \cdot 2^2 \]
No clear way makes \( 33^x = 324 \) straightforward with integers from provided options.
Given the available options, if they might relate to a mathematical task that is simpler:
Please check if there's a context for those responses since the equation \( 33^x = 324 \) doesn't yield a reasonable integer exponent among those values provided. If you have a specific context from which those numbers are drawn, please let me know!
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