Answer :
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].
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].