Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Join our interactive Q&A community and access a wealth of reliable answers to your most pressing questions.
Sagot :
To solve the equation [tex]\(5^{x-2} = 8\)[/tex] using logarithms and the change of base formula, follow these steps:
1. Take the logarithm of both sides of the equation: This allows us to bring the exponent down and make use of the logarithmic properties.
[tex]\[ \log(5^{x-2}) = \log(8) \][/tex]
2. Use the power rule of logarithms: The power rule states that [tex]\(\log(a^b) = b \log(a)\)[/tex]. Apply this to the left-hand side of the equation.
[tex]\[ (x-2) \log(5) = \log(8) \][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]: To solve for [tex]\(x\)[/tex], we need to isolate it. Start by dividing both sides of the equation by [tex]\(\log(5)\)[/tex].
[tex]\[ x - 2 = \frac{\log(8)}{\log(5)} \][/tex]
4. Solve for [tex]\(x\)[/tex]: Add 2 to both sides to isolate [tex]\(x\)[/tex].
[tex]\[ x = \frac{\log(8)}{\log(5)} + 2 \][/tex]
5. Calculate the logarithms and the final value of [tex]\(x\)[/tex]:
- [tex]\(\log(8) \approx 2.079\)[/tex]
- [tex]\(\log(5) \approx 1.609\)[/tex]
- So,
[tex]\[ x = \frac{2.079}{1.609} + 2 \approx 1.292 + 2 = 3.292 \][/tex]
6. Round the answer to the nearest thousandth:
[tex]\[ x \approx 3.292 \][/tex]
Therefore, the solution to the equation [tex]\(5^{x-2} = 8\)[/tex] is [tex]\(x \approx 3.292\)[/tex] when rounded to the nearest thousandth.
1. Take the logarithm of both sides of the equation: This allows us to bring the exponent down and make use of the logarithmic properties.
[tex]\[ \log(5^{x-2}) = \log(8) \][/tex]
2. Use the power rule of logarithms: The power rule states that [tex]\(\log(a^b) = b \log(a)\)[/tex]. Apply this to the left-hand side of the equation.
[tex]\[ (x-2) \log(5) = \log(8) \][/tex]
3. Isolate the term involving [tex]\(x\)[/tex]: To solve for [tex]\(x\)[/tex], we need to isolate it. Start by dividing both sides of the equation by [tex]\(\log(5)\)[/tex].
[tex]\[ x - 2 = \frac{\log(8)}{\log(5)} \][/tex]
4. Solve for [tex]\(x\)[/tex]: Add 2 to both sides to isolate [tex]\(x\)[/tex].
[tex]\[ x = \frac{\log(8)}{\log(5)} + 2 \][/tex]
5. Calculate the logarithms and the final value of [tex]\(x\)[/tex]:
- [tex]\(\log(8) \approx 2.079\)[/tex]
- [tex]\(\log(5) \approx 1.609\)[/tex]
- So,
[tex]\[ x = \frac{2.079}{1.609} + 2 \approx 1.292 + 2 = 3.292 \][/tex]
6. Round the answer to the nearest thousandth:
[tex]\[ x \approx 3.292 \][/tex]
Therefore, the solution to the equation [tex]\(5^{x-2} = 8\)[/tex] is [tex]\(x \approx 3.292\)[/tex] when rounded to the nearest thousandth.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.