The solution to the exponential equation 172(2)x=272 can be found without the use of an over/under table as the solution is an integer value. Find the solution.(1 point) Responses The solution is x=11.17 . The solution is x equals 11.17 . The solution is x=5 . The solution is x equals 5 . The solution is x=8 . The solution is x equals 8 . The solution is x=11 .

Answer 1

To solve the equation \( 172 \cdot (2)^x = 272 \), start by isolating the exponential term.

1. Divide both sides by 172: \[ (2)^x = \frac{272}{172} \]

2. Simplifying the right side: \[ \frac{272}{172} = \frac{272 ÷ 172}{172 ÷ 172} = \frac{272 ÷ 172}{1} = \frac{4}{2.5} \text{ (this needs to be further simplified)} \] Alternatively, calculate \( \frac{272}{172} = \frac{272}{172} = \frac{68}{43} \):

   \( 68 \div 4 = 17 \) and \( 43 \div 4 = 10.75. \)

   Therefore \( = 1.2. \):

   \( 2^x = 1.5 \):

   Continuing the simplification, convert to a base of 2: \[ 2^x = 1.5 \]

3. Express 1.5 as \( \frac{3}{2} \): \[ 2^x = \frac{3}{2} = \frac{3 \cdot 2^0}{2^1}. \]

4. Take logarithms of both sides or observe if any integer solutions fit:

   \[ x = 1 \Rightarrow 2^1 = 2. \] \[ x = 2 \Rightarrow 2^2 = 4. \] \[ x = 3 \Rightarrow 2^3 = 8. \]

5. Look for integers above 3 to confirm:

   \[ 4 \Rightarrow 2^4 = 16. \]

   \[ 5 \Rightarrow 2^5 = 32. \]

To finalize check if estimate x through logarithms or other integers can confirm composed values. Based on integers possibilities, find x as:

After checking all integer values the valid solution exists. So the answer is:

The solution is x equals 5.