Math Problem Statement

Find the value using log and antilog table: 3÷.82 ÷[(4.347 x √.06063) ÷ (.8721)²]

Solution

To solve this problem using logarithms and antilogarithms, follow these steps:

Problem:

Find the value of:

30.82÷[4.347×0.06063(0.8721)2]\frac{3}{0.82} \div \left[ \frac{4.347 \times \sqrt{0.06063}}{(0.8721)^2} \right]

Step-by-Step Solution:

  1. Simplify the Expression: First, break down the problem into smaller steps: Numerator: 30.82\text{Numerator: } \frac{3}{0.82} Denominator: 4.347×0.06063(0.8721)2\text{Denominator: } \frac{4.347 \times \sqrt{0.06063}}{(0.8721)^2}

  2. Logarithms: Use the logarithm properties to compute the values:

    Numerator: log10(30.82)\log_{10} \left( \frac{3}{0.82} \right)

    • log10(3)=0.4771\log_{10}(3) = 0.4771
    • log10(0.82)=0.0864\log_{10}(0.82) = -0.0864

    Using subtraction for division: log10(30.82)=0.4771(0.0864)=0.5635\log_{10} \left( \frac{3}{0.82} \right) = 0.4771 - (-0.0864) = 0.5635

    Denominator:

    log10[4.347×0.06063(0.8721)2]\log_{10} \left[ \frac{4.347 \times \sqrt{0.06063}}{(0.8721)^2} \right]

    • log10(4.347)=0.6382\log_{10}(4.347) = 0.6382

    • For 0.06063\sqrt{0.06063}, compute log10(0.06063)\log_{10}(0.06063): log10(0.06063)=1.2171\log_{10}(0.06063) = -1.2171 Divide by 2 (square root): 1.21712=0.6086\frac{-1.2171}{2} = -0.6086

    • Add log10(4.347)\log_{10}(4.347) and log10(0.06063)\log_{10}(\sqrt{0.06063}): 0.6382+(0.6086)=0.02960.6382 + (-0.6086) = 0.0296

    • For (0.8721)2(0.8721)^2, compute: log10(0.8721)=0.0594\log_{10}(0.8721) = -0.0594 Multiply by 2: 2×0.0594=0.11882 \times -0.0594 = -0.1188

    Subtract for the division: log10[4.347×0.06063(0.8721)2]=0.0296(0.1188)=0.1484\log_{10} \left[ \frac{4.347 \times \sqrt{0.06063}}{(0.8721)^2} \right] = 0.0296 - (-0.1188) = 0.1484

  3. Combine Numerator and Denominator: Subtract the denominator from the numerator: log10[30.824.347×0.06063(0.8721)2]=0.56350.1484=0.4151\log_{10} \left[ \frac{\frac{3}{0.82}}{\frac{4.347 \times \sqrt{0.06063}}{(0.8721)^2}} \right] = 0.5635 - 0.1484 = 0.4151

  4. Antilogarithm: Find the antilog of 0.41510.4151: 100.41512.60510^{0.4151} \approx 2.605

Final Answer:

2.605\boxed{2.605}

Let me know if you need a breakdown of any specific step or further clarification.


Related Questions:

  1. How do you simplify expressions involving logarithms and antilogarithms?
  2. What is the general formula for calculating square roots using logarithms?
  3. How can you find the antilogarithm without a calculator?
  4. Why is logarithmic computation useful for complex expressions?
  5. How do you interpret negative values in logarithmic calculations?

Tip:

Always double-check logarithm and antilogarithm values in the table to ensure precision.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Antilogarithms
Arithmetic Operations
Square Roots

Formulas

Logarithmic Property: log(a/b) = log(a) - log(b)
Logarithmic Property for Exponents: log(a^b) = b × log(a)
Square Root Using Logarithms: log(√a) = log(a)/2
Antilogarithm: antilog(log(a)) = a

Theorems

Logarithm Laws

Suitable Grade Level

Grades 10-12