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Using the formula, the decimal base logarithm is as follows:

10log x = n means x = 10n

where:

  • 10 is called the decimal base.
  • x is called logarithmic number or numerus with x> 0.
  • n is called the logarithm, power, or exponent of the decimal base.

Decimal base logarithm is the common (general) logarithm applied in the business, and is called logarithm (without decimal base).

Logarithms may be used to multiply or divide two numbers but are most useful in raising a number to a power, or extracting a (square, cube, etc) root of a number.

The logarithm (or log) of a number is denoted by expression “log ()”. Thus, the log of 3 is denoted by log (3).

There are some rules to know about logs:

  • There is no log for zero or negative numbers.
  • Since it is a decimal base, log (1)= Log (100)=0, log (10)= log (101)=1, log (100)=log (102)=2, and log (1000)=log (103)=3.
  • Numbers between 0 and 1 have negative logs.
  • Numbers between 1 and 10 have positive logs less than 1.
  • Numbers that are greater than 10 have positive logs and greater than 1.

To find the log of a number, we need to know:

  1. The position of the decimal point in the number (to find the characteristic or master number)
  2. The digits comprising the number (to find the mantissa).

All logs consist of a characteristic and a mantissa.

Steps to find the logs

  1. Remove the decimal point from the number, and, from what is left, find the mantissa from a log-table. Thus, to find log (303.00), we find the mantissa for the number 30300 as follows:
    Find 30 on the column N, and pull to the right until column 3 (or 300), we get 48144.
  2. Noting the position of the decimal point, use this rule to find the characteristic:

    1. If a number is greater than 1, count the number of digits to the left of decimal point. The characteristic is one less than this count and add 10 to this count.
    2. If the number is less than 1, count the number of zeroes between the decimal point and the first significant digit. Add 1 to this count, and place a negative sign before it. Finally, add 10 to this result. This is the characteristic.

With the characteristic and mantissa for a number, we write down the log using the following rule: begin by writing down the characteristic, place a decimal point, and write the mantissa after the decimal point. Next, subtract the result by 10. See the examples bellow:

  1. The log (303) will be written down as follows:
    The characteristic of 303.00 is 12 (using rule 2a: the number of digits to the left of decimal point=3 and the characteristic is one less than this count = 2, and add 10 equal to 12). The mantissa is 48144. Writing down the characteristic, place a decimal point, and write the mantissa after the decimal point, we get 12.48144. Subtract the result by 10: 12.48144-10=2.48144.
  2. The log (30.3) will be written down as follows:
    The characteristic of 30.3 is 11 (using rule 2a: the number of digits to the left of decimal point=2 and the characteristic is one less than this count = 1, and add 10 equal to 11). The mantissa is 48144. Writing down the characteristic, place a decimal point, and write the mantissa after the decimal point, we get 11.48144. Subtract the result by 10: 11.48144-10=1.48144.
  3. The log (3.03) will be written down as follows:
    The characteristic of 3.03 is 10 (using rule 2a: the number of digits to the left of decimal point=1 and the characteristic is one less than this count = 0, and add 10 equal to 10). The mantissa is 48144. Writing down the characteristic, place a decimal point, and write the mantissa after the decimal point, we get 10.48144. Subtract the result by 10: 10.48144-10=0.48144. 
  4. The log (0.303) will be written down as follows:
    The characteristic of 0.303 is 9 (using rule 2b: count the number of zeroes between the decimal point and the first significant digit (=0). Add 1 to this count (=1), and place a negative sign before it (=-1) Finally, add 10 to this result:-1+10=9). The mantissa is 48144. Writing down the characteristic, place a decimal point, and write the mantissa after the decimal point, we get 9.48144. Subtract the result by 10:
    9.48144-10=-0.51856.
    The log(0.303) can also be generated using the following operation:

    log (0.303)  = log (303/1000)
                       =  log (303) – log (1000)
                       = 2.48144 – 3
                       = -0.51856

  5. The log (0.0303) will be written down as follows:
    The characteristic of 0.0303 is 8 (using rule 2b: count the number of zeroes between the decimal pont and the first significant digit (=1). Add 1 to this count (=2), and place a negative sign before it (=-2) Finally, add 10 to this result:-2+10=8). The mantissa is 48144. Writing down the characteristic, place a decimal point, and write the mantissa after the decimal point, we get 8.48144. Subtract the result by 10: 8.48144-10=-1. 51856.

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