Answer:
2495
Explanation:
You want to evaluate 61.73² -36.27² using log tables.
Logs
Logarithms are useful for changing a multiplication problem to an addition problem. They cannot be used to find a sum, unless that sum can be expressed as a product.
Difference of squares
We know the difference of squares can be rearranged to the product ...
a² -b² = (a -b)(a +b)
For the expression of interest, this means ...
61.73² -36.27² = (61.73 -36.27)(61.73 +36.27) = 25.46×98
With numbers
The log of the product is the sum of the logs of the factors:
log(25.46×98) = log(25.46) +log(98)
log(25.46×98) = (1 +0.405858) +(1 +0.991226) = 3.397084
We want the number that has this logarithm, so we use the table in reverse:
25.46×98 ≈ 2.495×10³ ≈ 2495
The difference of interest is about 2495.
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Additional comment
We have assumed a log table that can be interpolated to give 6 significant figures of logarithm, and 4 significant figures of antilog. If your table is more extensive than that, you may be able to obtain a result with more precision.
The actual difference is 2495.08. The antilog of our logarithm is about 2495.077, so rounds to the same value.
You could evaluate the squares separately, then find their difference. That requires another antilog lookup in the middle of the calculation, and tends to reduce the accuracy. 10^(2·log(61.73)) ≈ 3811, 10^(2·log(36.27)) ≈ 1316, so the difference is 3811 -1316 = 2495.
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