Math Problem Statement

Evaluate the expression \( \sqrt{\frac{0.4271 \times 609.3}{86.08^2}} \) using logarithms only.

Solution

From the image, it seems you're tasked with solving a mathematical problem using logarithms. The problem appears to be:

0.4271×609.386.082\sqrt{\frac{0.4271 \times 609.3}{86.08^2}}

and the instruction says to "use logarithms only" to evaluate this.

Let me guide you through the solution using logarithms:


Step 1: Use the logarithm property for products, quotients, and powers.

We aim to simplify the expression inside the square root using logarithmic properties.

Expression: 0.4271×609.386.082\text{Expression: } \sqrt{\frac{0.4271 \times 609.3}{86.08^2}}

Taking the logarithm (logarithm base 10, denoted as log\log):

log(Expression)=12log(0.4271×609.386.082)\log(\text{Expression}) = \frac{1}{2} \log\left(\frac{0.4271 \times 609.3}{86.08^2}\right)

Using the logarithmic property log(A/B)=log(A)log(B)\log(A/B) = \log(A) - \log(B):

log(Expression)=12[log(0.4271×609.3)log(86.082)]\log(\text{Expression}) = \frac{1}{2} \left[\log(0.4271 \times 609.3) - \log(86.08^2)\right]


Step 2: Expand using log(A×B)=log(A)+log(B)\log(A \times B) = \log(A) + \log(B) and log(A2)=2log(A)\log(A^2) = 2\log(A).

log(Expression)=12[log(0.4271)+log(609.3)2log(86.08)]\log(\text{Expression}) = \frac{1}{2} \left[\log(0.4271) + \log(609.3) - 2\log(86.08)\right]


Step 3: Calculate each term using logarithmic tables or a calculator.

  1. log(0.4271)0.3693\log(0.4271) \approx -0.3693
  2. log(609.3)2.7842\log(609.3) \approx 2.7842
  3. log(86.08)1.9342\log(86.08) \approx 1.9342

Substitute these values into the equation:

log(Expression)=12[0.3693+2.78422(1.9342)]\log(\text{Expression}) = \frac{1}{2} \left[-0.3693 + 2.7842 - 2(1.9342)\right]


Step 4: Simplify inside the brackets.

0.3693+2.7842=2.4149-0.3693 + 2.7842 = 2.4149 2.41492(1.9342)=2.41493.8684=1.45352.4149 - 2(1.9342) = 2.4149 - 3.8684 = -1.4535

So: log(Expression)=12(1.4535)=0.7268\log(\text{Expression}) = \frac{1}{2}(-1.4535) = -0.7268


Step 5: Find the antilog to get the value of the expression.

To get the value of the square root expression, take the antilog of 0.7268-0.7268:

Expression=100.72680.1878\text{Expression} = 10^{-0.7268} \approx 0.1878


Final Answer:

0.4271×609.386.0820.188\sqrt{\frac{0.4271 \times 609.3}{86.08^2}} \approx 0.188

Let me know if you need further clarifications or detailed explanations.


5 Related Questions:

  1. What are the key logarithmic properties used in solving such problems?
  2. How do you calculate the antilog of a given value manually?
  3. Can this problem be solved without logarithms? If so, how?
  4. Why do we divide by 2 when the square root is involved in logarithmic operations?
  5. How do logarithms simplify complex mathematical operations?

Tip:

Always remember to keep a consistent number of significant figures throughout your calculations for precision.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Square Roots
Exponents
Arithmetic Simplification

Formulas

Logarithm product rule: \( \log(A \times B) = \log(A) + \log(B) \)
Logarithm quotient rule: \( \log(A / B) = \log(A) - \log(B) \)
Logarithm power rule: \( \log(A^n) = n \log(A) \)
Square root property: \( \sqrt{x} = x^{1/2} \)

Theorems

Logarithmic Rules
Exponent Rules

Suitable Grade Level

Grades 10-12