Question

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=8
.
The solution is x equals 8 .

The solution is x=5
.
The solution is x equals 5 .

The solution is x=11
.
The solution is x equals 11 .

The solution is x=11.17
.
The solution is x equals 11.17 .

Answer 1

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

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

2. Simplify \( \frac{272}{172} \):
\[
\frac{272}{172} = \frac{272 \div 28}{172 \div 28} = \frac{68}{43}
\]

3. 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:
\[
\text{The solution is } x = 8.
\]