Answer :
Certainly! Let's work through the expression step by step and use the logarithm table to evaluate it:
Given expression:
[tex]\[ \left[\frac{0.009532}{9.134}\right] \times \frac{3 \sqrt{48.27}}{403.2} \][/tex]
### Step 1: Simplify Each Fraction
First Fraction:
[tex]\[ \frac{0.009532}{9.134} \][/tex]
Using the logarithm table, we find the logarithms of the numerator and the denominator:
- [tex]\(\log_{10}(0.009532) \approx -2.0212\)[/tex]
- [tex]\(\log_{10}(9.134) \approx 0.9606\)[/tex]
Now, subtract the logarithms to find the logarithm of the fraction:
[tex]\[ \log_{10}\left(\frac{0.009532}{9.134}\right) = \log_{10}(0.009532) - \log_{10}(9.134) \approx -2.0212 - 0.9606 = -2.9818 \][/tex]
Converting this back from logarithms:
[tex]\[ \frac{0.009532}{9.134} \approx 10^{-2.9818} \approx 0.0010435734617911102 \][/tex]
Second Fraction:
[tex]\[ \frac{3 \sqrt{48.27}}{403.2} \][/tex]
First, calculate [tex]\(\sqrt{48.27}\)[/tex] using logarithms:
- [tex]\(\log_{10}(48.27) \approx 1.6838\)[/tex]
Using the property of logarithms:
[tex]\[ \log_{10}(\sqrt{48.27}) = \frac{1}{2} \log_{10}(48.27) \approx \frac{1}{2} \times 1.6838 = 0.8419 \][/tex]
Convert back from logarithms:
[tex]\[ \sqrt{48.27} \approx 10^{0.8419} \approx 6.92 \][/tex]
Now, calculate the logarithm of the numerator [tex]\(3 \times 6.92\)[/tex]:
- [tex]\(\log_{10}(3) \approx 0.4771\)[/tex]
- [tex]\(\log_{10}(6.92) \approx 0.8419\)[/tex]
[tex]\[ \log_{10}(3 \times 6.92) = \log_{10}(3) + \log_{10}(6.92) \approx 0.4771 + 0.8419 = 1.319 \][/tex]
Convert back from logarithms:
[tex]\[ 3 \sqrt{48.27} \approx 10^{1.319} \approx 20.89 \][/tex]
Now, evaluate the second fraction:
[tex]\[ \frac{20.89}{403.2} \][/tex]
Using the logarithm table:
- [tex]\(\log_{10}(20.89) \approx 1.319\)[/tex]
- [tex]\(\log_{10}(403.2) \approx 2.6053\)[/tex]
[tex]\[ \log_{10}\left(\frac{20.89}{403.2}\right) = \log_{10}(20.89) - \log_{10}(403.2) \approx 1.319 - 2.6053 = -1.2863 \][/tex]
Convert back from logarithms:
[tex]\[ \frac{3 \sqrt{48.27}}{403.2} \approx 10^{-1.2863} \approx 0.051693909799648066 \][/tex]
### Step 2: Combine the Fractions
Now, multiply these two results:
[tex]\[ \left[\frac{0.009532}{9.134}\right] \times \frac{3 \sqrt{48.27}}{403.2} = 0.0010435734617911102 \times 0.051693909799648066 \][/tex]
Using the logarithm table:
- [tex]\(\log_{10}(0.0010435734617911102) \approx -2.9818\)[/tex]
- [tex]\(\log_{10}(0.051693909799648066) \approx -1.2863\)[/tex]
[tex]\[ \log_{10}(0.0010435734617911102 \times 0.051693909799648066) = -2.9818 + -1.2863 = -4.2681 \][/tex]
Convert back from logarithms:
[tex]\[ 0.0010435734617911102 \times 0.051693909799648066 \approx 10^{-4.2681} \approx 5.394639240313613 \times 10^{-5} \][/tex]
Thus, the evaluated result of the given expression is:
[tex]\[ \boxed{5.394639240313613 \times 10^{-5}} \][/tex]
Given expression:
[tex]\[ \left[\frac{0.009532}{9.134}\right] \times \frac{3 \sqrt{48.27}}{403.2} \][/tex]
### Step 1: Simplify Each Fraction
First Fraction:
[tex]\[ \frac{0.009532}{9.134} \][/tex]
Using the logarithm table, we find the logarithms of the numerator and the denominator:
- [tex]\(\log_{10}(0.009532) \approx -2.0212\)[/tex]
- [tex]\(\log_{10}(9.134) \approx 0.9606\)[/tex]
Now, subtract the logarithms to find the logarithm of the fraction:
[tex]\[ \log_{10}\left(\frac{0.009532}{9.134}\right) = \log_{10}(0.009532) - \log_{10}(9.134) \approx -2.0212 - 0.9606 = -2.9818 \][/tex]
Converting this back from logarithms:
[tex]\[ \frac{0.009532}{9.134} \approx 10^{-2.9818} \approx 0.0010435734617911102 \][/tex]
Second Fraction:
[tex]\[ \frac{3 \sqrt{48.27}}{403.2} \][/tex]
First, calculate [tex]\(\sqrt{48.27}\)[/tex] using logarithms:
- [tex]\(\log_{10}(48.27) \approx 1.6838\)[/tex]
Using the property of logarithms:
[tex]\[ \log_{10}(\sqrt{48.27}) = \frac{1}{2} \log_{10}(48.27) \approx \frac{1}{2} \times 1.6838 = 0.8419 \][/tex]
Convert back from logarithms:
[tex]\[ \sqrt{48.27} \approx 10^{0.8419} \approx 6.92 \][/tex]
Now, calculate the logarithm of the numerator [tex]\(3 \times 6.92\)[/tex]:
- [tex]\(\log_{10}(3) \approx 0.4771\)[/tex]
- [tex]\(\log_{10}(6.92) \approx 0.8419\)[/tex]
[tex]\[ \log_{10}(3 \times 6.92) = \log_{10}(3) + \log_{10}(6.92) \approx 0.4771 + 0.8419 = 1.319 \][/tex]
Convert back from logarithms:
[tex]\[ 3 \sqrt{48.27} \approx 10^{1.319} \approx 20.89 \][/tex]
Now, evaluate the second fraction:
[tex]\[ \frac{20.89}{403.2} \][/tex]
Using the logarithm table:
- [tex]\(\log_{10}(20.89) \approx 1.319\)[/tex]
- [tex]\(\log_{10}(403.2) \approx 2.6053\)[/tex]
[tex]\[ \log_{10}\left(\frac{20.89}{403.2}\right) = \log_{10}(20.89) - \log_{10}(403.2) \approx 1.319 - 2.6053 = -1.2863 \][/tex]
Convert back from logarithms:
[tex]\[ \frac{3 \sqrt{48.27}}{403.2} \approx 10^{-1.2863} \approx 0.051693909799648066 \][/tex]
### Step 2: Combine the Fractions
Now, multiply these two results:
[tex]\[ \left[\frac{0.009532}{9.134}\right] \times \frac{3 \sqrt{48.27}}{403.2} = 0.0010435734617911102 \times 0.051693909799648066 \][/tex]
Using the logarithm table:
- [tex]\(\log_{10}(0.0010435734617911102) \approx -2.9818\)[/tex]
- [tex]\(\log_{10}(0.051693909799648066) \approx -1.2863\)[/tex]
[tex]\[ \log_{10}(0.0010435734617911102 \times 0.051693909799648066) = -2.9818 + -1.2863 = -4.2681 \][/tex]
Convert back from logarithms:
[tex]\[ 0.0010435734617911102 \times 0.051693909799648066 \approx 10^{-4.2681} \approx 5.394639240313613 \times 10^{-5} \][/tex]
Thus, the evaluated result of the given expression is:
[tex]\[ \boxed{5.394639240313613 \times 10^{-5}} \][/tex]
