Question

What exponent would make the two expressions equivalent: (33)x=324?

15
15

21
21

8
8

27

Answer 1

To find the exponent \( x \) that makes the two expressions \( (33)^x = 324 \) equivalent, we need to solve for \( x \).

First, we can rewrite \( 324 \) as a power of \( 33 \).

We can also rewrite \( 33 \) as a power base that could help us find if \( 324 \) can be expressed in a similar way. However, a more direct approach involves using logarithms.

Taking the logarithm of both sides, we can write:

\[
x \cdot \log(33) = \log(324)
\]

Then, solving for \( x \):

\[
x = \frac{\log(324)}{\log(33)}
\]

Now, we can calculate \( \log(324) \) and \( \log(33) \).

For \( \log(324) \):

\[
324 = 18^2 = (2 \cdot 3^2)^2 = 2^2 \cdot 3^4
\]

Alternatively, directly, we can compute \( \log(324) \):

Finding \( \log(33) \), we can evaluate \( x \):

Using a calculator or log tables:

\[
\log(324) \approx 2.510
\]
\[
\log(33) \approx 1.519
\]

Substituting the logs into our equation:

\[
x \approx \frac{2.510}{1.519} \approx 1.65
\]

However, since this value does not align with the choices you provided (15, 21, 8, 27), let’s check if any of these values satisfy \( (33)^x = 324 \):

- Calculating \( (33)^8 \) is enormous compared to \( 324 \).
- \( (33)^{15} \) is also far too large.
- \( (33)^{21} \) does not match \( 324 \).
- \( (33)^{27} \) is similarly too large.

To find whether any choice actually satisfies \( (33)^x = 324 \), we can approximate and test or calculate.

At a glance, none of the offered choices appear to be feasible exponents corresponding to \( 324 \).

From these testings, none of the answers provided seem accurate without context or additional insight, or perhaps there's a mistake in anticipated answers.

Thus, unless using computational checks to thoroughly assess potential errors or filings, confirming your selections against exponential growth rate behaviors, the steps I outlined above reflect upon log evaluations for perspective checking with the mapped out constraints.