Answer:
197.2
Explanation:
You want to use a table of logarithms to evaluate the product 13.81×14.28.
Logarithms
Logarithms turn a multiplication problem into an addition problem, along with some table look-ups. The basic idea is ...
log(ab) = log(a) +log(b)
Finding the log
To find the logarithm of a number, it can help to write it in scientific notation. This puts one non-zero digit to the left of the decimal point and expresses the multiplier as a power of 10. The first attachment shows a number in scientific notation.
A log table is used to find the logarithm of the mantissa of the number in scientific notation. The result will be a decimal fraction between 0 and 1.
You start by locating the first digits of the mantissa in the left column of the table, and additional digits across the top. The second attachment shows a portion of a table of logarithms with the relevant table entries identified.
The log of 13.81 = 1.381×10¹ is found by locating 13 in the left column, and the next digit, 8, at the top of the table. The final digit, 1, is located at the top of the portion of the table labeled Mean Difference. The number in the Mean Difference column is added to the number found in row 13, column 8: 1399 +3 = 1402. This sum is the mantissa of the logarithm of 13.81. (See the third attachment for the naming we're using.)
Similarly, the log of 14.28 is found in row 14, column 2, and the value in Mean Difference column 8 is added to that: 1523 +24 = 1547.
The characteristic of each logarithm is the corresponding characteristic of the number in scientific notation. For each of these numbers, it is 1.
- log(1.381×10¹) = 1.1402
- log(1.428×10¹) = 1.1547
Log of product
The logarithm of the product is the sum of these logarithms:
log(1.381×10¹ · 1.428×10¹) = 1.1402 +1.1547 = 2.2949
Finding the antilog
The value corresponding to this logarithm is found by a procedure that is the reverse of the lookup procedure.
Our logarithm has a characteristic of 2, and a mantissa of 0.2949.
First, we locate the highest number in the table that is less than or equal to the mantissa. Here, that value is 2945, located in row 19, column 7 of the table. The value we want, 2949, is 4 more than this, so we look in the Mean Difference table on row 19 for a value of 4. That is found in column 2.
The mantissa of the antilog of 2949 is the row number (19) with the column numbers appended: 1972. The decimal point for this is located to the right of the leading digit: 1.972.
The characteristic of the antilog is the characteristic of the logarithm: 2.
Then the antilog in scientific notation is ...
10^2.2949 = 1.972×10²
In standard form, this is ...
1.972×10² = 197.2
Product
The desired product is found to be 13.81×14.28 ≈ 197.2.
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Additional comment
Different tables of logs may show the interpolation values (Mean Difference) differently. These values are intended to aid in interpolating between table values so that you can find logs and antilogs of numbers with 4 significant digits. (Larger tables let you work with more significant digits.)
In general, the number of significant digits in the values you work with will be less than or equal to the number of significant digits in the logs in the table. Here, the table has 4-digit logs, so our result, 197.2, can have at most 4 significant digits. (The actual product is 197.2068.)
When the characteristic of a number is negative, care must be taken to use correct values for the mantissa and characteristic of the logarithm. Table values are logs of numbers between 1 and 10, and are positive. The log of 0.003949 would be the sum of the negative characteristic -3 and the positive logarithm mantissa 0.5965. That sum would be -2.4035 = (-2) + (-0.4035). Since we want to keep the fractional part positive, we sometimes add and subtract a large positive number. The logarithm of 0.003949 might be expressed in this way as 7.5965-10, effectively the log of 3.949×10⁷×10⁻¹⁰.