To solve the exponential equation \( 172(2)^x = 272 \), we can start by isolating \( (2)^x \).
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Divide both sides by 172: \[ (2)^x = \frac{272}{172} \]
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Simplify \( \frac{272}{172} \): \[ \frac{272}{172} = \frac{272 \div 28}{172 \div 28} = \frac{68}{43} \]
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Re-write the equation: \[ (2)^x = \frac{68}{43} \]
Next, we can rewrite 68 and 43 as powers of 2 and solve for x:
- 68 can be approximated as \( 2^6 \) (since \( 2^6 = 64 \)) and is roughly equivalent to it.
- 43 can be approximated as \( 2^5 \).
Next step is to express our fraction in terms of base 2: \[ (2)^x \approx 1.581 \]
To further simplify: You can also find common denominators or use logs for better approximation, but we keep this as:
By trial of logarithms or evaluating integers, we would quickly find suitable integer \( x\) around given choice if exact powers were not a solution.
Ultimately, checking values based will yield:
- \( 2^5 = 32\) which is too low
- \( 2^6 = 64 \)
- \( 2^7 = 128 \)
- \( 2^{8} = 256\) bigger than our target.
Thus it becomes clear \( x = 8 \).
Thus, the solution is: \[ \text{The solution is } x = 8. \]