To solve the inequality [tex]\(2^{x-7} \geq 3\)[/tex], we need to isolate [tex]\(x\)[/tex]. Here's a step-by-step explanation:
1. Start with the given inequality:
[tex]\[
2^{x-7} \geq 3
\][/tex]
2. Take the natural logarithm (ln) of both sides:
[tex]\[
\ln(2^{x-7}) \geq \ln(3)
\][/tex]
3. Use the property of logarithms to bring down the exponent:
[tex]\[
(x-7) \cdot \ln(2) \geq \ln(3)
\][/tex]
4. Isolate [tex]\(x\)[/tex] by dividing both sides by [tex]\(\ln(2)\)[/tex]:
[tex]\[
x - 7 \geq \frac{\ln(3)}{\ln(2)}
\][/tex]
5. Calculate the values of [tex]\(\ln(3)\)[/tex] and [tex]\(\ln(2)\)[/tex]:
[tex]\[
\ln(3) \approx 1.0986122886681098
\][/tex]
[tex]\[
\ln(2) \approx 0.6931471805599453
\][/tex]
6. Calculate [tex]\(\frac{\ln(3)}{\ln(2)}\)[/tex]:
[tex]\[
\frac{\ln(3)}{\ln(2)} \approx \frac{1.0986122886681098}{0.6931471805599453} \approx 1.5849625007211563
\][/tex]
7. Solve for [tex]\(x\)[/tex]:
[tex]\[
x - 7 \geq 1.5849625007211563
\][/tex]
8. Add 7 to both sides:
[tex]\[
x \geq 8.584962500721156
\][/tex]
9. Determine the least integer value of [tex]\(x\)[/tex] that satisfies the inequality:
[tex]\[
\lceil 8.584962500721156 \rceil = 9
\][/tex]
Thus, the least integer value that satisfies the inequality [tex]\(2^{x-7} \geq 3\)[/tex] is [tex]\( \boxed{9} \)[/tex].
