Find the value using Logarithm Table. 31.3 74.3 47.3

To find the value of $\frac{52.23\times 15.4}{69.35\times 2.76}$ using logarithm tables, we will use the properties of logarithms to simplify the calculation. The steps are as follows:

Step 1: Express the given expression in logarithmic form

We can rewrite the expression using the logarithm properties: $$\log \left(\frac{52.23\times 15.4}{69.35\times 2.76}\right)=\log (52.23)+\log (15.4)-\log (69.35)-\log (2.76)$$

Step 2: Find the logarithms of each number using the logarithm table

Assume the following values from the logarithm table (rounded to 4 decimal places for simplicity):

• $\log (52.23)=1.7179$
• $\log (15.4)=1.1875$
• $\log (69.35)=1.8411$
• $\log (2.76)=0.4409$

Step 3: Substitute the values into the logarithmic expression

$$\log \left(\frac{52.23\times 15.4}{69.35\times 2.76}\right)=1.7179+1.1875-1.8411-0.4409$$

Step 4: Perform the arithmetic operations

$$1.7179+1.1875=2.9054$$ $$1.8411+0.4409=2.2820$$ $$2.9054-2.2820=0.6234$$

Step 5: Find the antilogarithm of the result

The result of the logarithmic calculation is $0.6234$. To find the original value, we take the antilogarithm of $0.6234$:

$$\text{Antilog}(0.6234)\approx 4.20$$

Thus, the value of the given expression is approximately 4.20.

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Detailed Breakdown of Logarithmic Calculation

For clarity, let's break down the calculation further:

1. Logarithm of numerator terms:

   • $\log (52.23)=1.7179$
   • $\log (15.4)=1.1875$
   • Sum: $1.7179+1.1875=2.9054$

2. Logarithm of denominator terms:

   • $\log (69.35)=1.8411$
   • $\log (2.76)=0.4409$
   • Sum: $1.8411+0.4409=2.2820$

3. Subtract denominator sum from numerator sum: $$2.9054-2.2820=0.6234$$

4. Convert back from logarithm to actual value:

   • The characteristic (integer part) of $0.6234$ is 0, and the mantissa (decimal part) is 0.6234.
   • Using an antilogarithm table or calculator, we find: $$\text{Antilog}(0.6234)\approx 4.20$$

Verification

To ensure accuracy, let's verify the calculation using direct computation:

$$\frac{52.23\times 15.4}{69.35\times 2.76}=\frac{805.742}{191.361}\approx 4.21$$

The slight difference (4.20 vs. 4.21) is due to rounding errors in the logarithm table values. For the purpose of this problem, 4.20 is the acceptable answer based on the logarithmic method.