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 follow these steps:

Step 1: Break down the expression using logarithm properties

We use the following logarithm properties:

1. $\log (ab)=\log a+\log b$
2. $\log \left(\frac{a}{b}\right)=\log a-\log b$

Thus, the expression can be rewritten as: $$\log \left(\frac{52.23\times 15.4}{69.35\times 2.76}\right)=\log (52.23)+\log (15.4)-\log (69.35)-\log (2.76)$$

Step 2: Find the logarithm values from the table

Using the logarithm table (base 10), we find the following values:

• $\log (52.23)$:

  • $\log (52)=1.7160$
  • For the remaining part (0.23), we use the mean difference table:
    • Mean difference for 23 is approximately 0.0092.
  • Thus, $\log (52.23)=1.7160+0.0092=1.7252$

• $\log (15.4)$:

  • $\log (15)=1.1761$
  • For the remaining part (0.4), we use the mean difference table:
    • Mean difference for 4 is approximately 0.0061.
  • Thus, $\log (15.4)=1.1761+0.0061=1.1822$

• $\log (69.35)$:

  • $\log (69)=1.8388$
  • For the remaining part (0.35), we use the mean difference table:
    • Mean difference for 35 is approximately 0.0136.
  • Thus, $\log (69.35)=1.8388+0.0136=1.8524$

• $\log (2.76)$:

  • $\log (2.7)=0.4314$
  • For the remaining part (0.06), we use the mean difference table:
    • Mean difference for 6 is approximately 0.0026.
  • Thus, $\log (2.76)=0.4314+0.0026=0.4340$

Step 3: Substitute the values into the expression

$$\log \left(\frac{52.23\times 15.4}{69.35\times 2.76}\right)=1.7252+1.1822-1.8524-0.4340$$

Step 4: Perform the arithmetic operations

$$1.7252+1.1822=2.9074$$ $$2.9074-1.8524=1.0550$$ $$1.0550-0.4340=0.6210$$

Step 5: Find the antilogarithm of the result

The result of the logarithm is 0.6210. To find the original value, we take the antilogarithm (base 10) of 0.6210.

Using the antilogarithm table:

• The characteristic (integer part) is 0, so the decimal part is 0.6210.
• The antilogarithm of 0.6210 is approximately 4.173.

Thus, the value of the expression is 4.173.