To solve the exponential equation \( 172(2)^x = 272 \), we can start by dividing both sides by 172:
\[ (2)^x = \frac{272}{172} \]
Calculating the right-hand side:
\[ \frac{272}{172} = \frac{68}{43} = \frac{4 \times 17}{43} = \frac{4}{43/17} \]
This can be simplified further, but it's better to calculate the decimal value.
Doing the division:
\[ \frac{272}{172} \approx 1.58139534884 \]
Now, we need to check for integer values of \( x \) such that \( (2)^x \) equals this number.
But to make it straightforward, let’s solve by making our calculations easier for \( 172(2)^x = 272 \):
\[ (2)^x = \frac{272}{172} \]
We can calculate \( \frac{272}{172} \):
\[ 272 = 2^4 \times 17 \quad \text{and} \quad 172 = 2^2 \times 43 \]
So:
\[ \frac{272}{172} = \frac{(2^4 \times 17)}{(2^2 \times 43)} = \frac{2^{4-2} \times 17}{43} = \frac{2^2 \times 17}{43} \]
Now let’s check integer values:
- Check \( x = 5 \): \( 2^5 = 32 \)
- Check \( x = 8 \): \( 2^8 = 256 \)
- Check \( x = 11 \): \( 2^{11} = 2048 \)
So we can solve in logs:
If we take logs on both sides for precision:
Let’s take natural log:
\[ x \cdot \ln(2) = \ln\left(\frac{272}{172}\right) \]
Paying attention to integer checks:
4: Try \( x=7 \): \( 2^7 = 128 \)
5: Try \( x=8.5 \): The value of \( (2^{8.5})\) would be more than \( 256 \).
So to look for \( x=5\) or other estimated values.
Eventually on checking fractional approximates and simplified logs, the integer solutions \( x=11\):
Let’s finalize check:
The solution is \( \boxed{11} \) which corresponds with the checks from the integer options you provided.