Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To evaluate the expression [tex]\(\frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}}\)[/tex] using logarithms, we can break it down into a series of steps. Here is a detailed, step-by-step solution:
1. Identify the components of the expression:
The given expression is:
[tex]\[ \frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}} \][/tex]
2. Use logarithms to simplify the expression:
We know that using logarithm properties can help simplify the calculations. Recall the properties:
[tex]\[ \log(a^b) = b \log(a) \][/tex]
and
[tex]\[ \log\left(\frac{a \cdot b}{c}\right) = \log(a) + \log(b) - \log(c) \][/tex]
3. Break down the expression using logarithms:
Let:
[tex]\[ a = 17.2, \quad b = 4.93, \quad c = 6750000, \quad \text{and} \quad d = \frac{1}{3} \][/tex]
We need to find:
[tex]\[ \log\left(\frac{17.2^2 \times 4.93}{6750000^{1/3}}\right) \][/tex]
Using the properties of logarithms:
[tex]\[ \log\left(17.2^2 \times 4.93\right) = \log(17.2^2) + \log(4.93) \][/tex]
So,
[tex]\[ \log(17.2^2) = 2 \log(17.2) \][/tex]
Now we rewrite the logarithm of the denominator using the property of exponents:
[tex]\[ \log(6750000^{1/3}) = \frac{1}{3} \log(6750000) \][/tex]
4. Calculate individual logarithms using values:
From the logarithm tables or given data:
[tex]\[ \log(17.2) = 1.235528446907549 \][/tex]
[tex]\[ \log(4.93) = 0.69284691927723 \][/tex]
[tex]\[ \log(6750000) = 6.829303772831025 \][/tex]
Therefore:
[tex]\[ 2 \log(17.2) = 2 \times 1.235528446907549 = 2.471056893815098 \][/tex]
[tex]\[ \frac{1}{3} \log(6750000) = \frac{1}{3} \times 6.829303772831025 = 2.276434590943675 \][/tex]
5. Combine the logarithmic terms:
Now, put these values into the expression:
[tex]\[ \log\left(\frac{17.2^2 \times 4.93}{6750000^{1/3}}\right) = 2.471056893815098 + 0.69284691927723 - 2.276434590943675 \][/tex]
Simplify the right side:
[tex]\[ 2.471056893815098 + 0.69284691927723 - 2.276434590943675 = 0.8874692221486531 \][/tex]
6. Convert back from logarithms to find the result:
To find the final result, convert [tex]\(\log(x) = 0.8874692221486531\)[/tex] back to [tex]\(x\)[/tex]:
[tex]\[ x = 10^{0.8874692221486531} \][/tex]
Therefore:
[tex]\[ x \approx 7.7173682172212095 \][/tex]
Hence, the value of [tex]\(\frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}}\)[/tex] evaluates to approximately [tex]\(7.7173682172212095\)[/tex].
1. Identify the components of the expression:
The given expression is:
[tex]\[ \frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}} \][/tex]
2. Use logarithms to simplify the expression:
We know that using logarithm properties can help simplify the calculations. Recall the properties:
[tex]\[ \log(a^b) = b \log(a) \][/tex]
and
[tex]\[ \log\left(\frac{a \cdot b}{c}\right) = \log(a) + \log(b) - \log(c) \][/tex]
3. Break down the expression using logarithms:
Let:
[tex]\[ a = 17.2, \quad b = 4.93, \quad c = 6750000, \quad \text{and} \quad d = \frac{1}{3} \][/tex]
We need to find:
[tex]\[ \log\left(\frac{17.2^2 \times 4.93}{6750000^{1/3}}\right) \][/tex]
Using the properties of logarithms:
[tex]\[ \log\left(17.2^2 \times 4.93\right) = \log(17.2^2) + \log(4.93) \][/tex]
So,
[tex]\[ \log(17.2^2) = 2 \log(17.2) \][/tex]
Now we rewrite the logarithm of the denominator using the property of exponents:
[tex]\[ \log(6750000^{1/3}) = \frac{1}{3} \log(6750000) \][/tex]
4. Calculate individual logarithms using values:
From the logarithm tables or given data:
[tex]\[ \log(17.2) = 1.235528446907549 \][/tex]
[tex]\[ \log(4.93) = 0.69284691927723 \][/tex]
[tex]\[ \log(6750000) = 6.829303772831025 \][/tex]
Therefore:
[tex]\[ 2 \log(17.2) = 2 \times 1.235528446907549 = 2.471056893815098 \][/tex]
[tex]\[ \frac{1}{3} \log(6750000) = \frac{1}{3} \times 6.829303772831025 = 2.276434590943675 \][/tex]
5. Combine the logarithmic terms:
Now, put these values into the expression:
[tex]\[ \log\left(\frac{17.2^2 \times 4.93}{6750000^{1/3}}\right) = 2.471056893815098 + 0.69284691927723 - 2.276434590943675 \][/tex]
Simplify the right side:
[tex]\[ 2.471056893815098 + 0.69284691927723 - 2.276434590943675 = 0.8874692221486531 \][/tex]
6. Convert back from logarithms to find the result:
To find the final result, convert [tex]\(\log(x) = 0.8874692221486531\)[/tex] back to [tex]\(x\)[/tex]:
[tex]\[ x = 10^{0.8874692221486531} \][/tex]
Therefore:
[tex]\[ x \approx 7.7173682172212095 \][/tex]
Hence, the value of [tex]\(\frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}}\)[/tex] evaluates to approximately [tex]\(7.7173682172212095\)[/tex].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
