14
Number Notations, Arithmetical Operations, and Calculators
notation and before pressing the log key if you wish to obtain the log to more
than four decimal places.
Because logarithms are exponents, we have the following logarithm laws that
are derived from the laws of exponents given on page 8. Let A and B be any
two numbers.
Log of a product:
logA5 = log A + log B
A
Log of a quotient:
log -=: = log A - log B
Log of a power (n):
log A" = n log A
Log of the n
th root:
log X/A = log A
1
"
1 = - log A
The logarithm of a number consists of two parts, called the characteristic and
the mantissa. The characteristic is the portion of the log that lies before the
decimal point, and the mantissa is the portion that lies after the decimal point.
The significance of separating a logarithm into these two parts is evident when
you apply the logarithm laws to the logs of numbers such as 2000, and 2, and
0.000002.
log 2000 = log (2 x 10
3
) = log 2 + log 10
3 = 0.30103 + 3 = 3.30103
log 2 = log (2 x 10°) = log 2 + log 10° = 0.30103 + 0 = 0.30103
log 0.000002 = log (2 x 108
) = log 2 + log 1Q6 = 0.30103 - 6 = -5.69897
Note that the characteristic is determined by the power to which 10 is raised
(when the number is in standard scientific notation), and the mantissa is determined by the log of the lefthand factor (when the number is in scientific notation). It is these properties that make it so easy to find the logarithm of a number
using a log table. Here is how you can do it.
1. Write the number (N) in standard scientific notation.
2. Look up the mantissa in the log table. It is the log of the lefthand factor
in scientific notation, which is a number between 1 and 10. The mantissa will lie between 0 and 1.
3. The exponent of 10 (the righthand factor) is the characteristic of the log.
4. Add the mantissa and the characteristic to obtain log N.
PROBLEM:
Find the log of 203.
Number Notations, Arithmetical Operations, and Calculators
notation and before pressing the log key if you wish to obtain the log to more
than four decimal places.
Because logarithms are exponents, we have the following logarithm laws that
are derived from the laws of exponents given on page 8. Let A and B be any
two numbers.
Log of a product:
logA5 = log A + log B
A
Log of a quotient:
log -=: = log A - log B
Log of a power (n):
log A" = n log A
Log of the n
th root:
log X/A = log A
1
"
1 = - log A
The logarithm of a number consists of two parts, called the characteristic and
the mantissa. The characteristic is the portion of the log that lies before the
decimal point, and the mantissa is the portion that lies after the decimal point.
The significance of separating a logarithm into these two parts is evident when
you apply the logarithm laws to the logs of numbers such as 2000, and 2, and
0.000002.
log 2000 = log (2 x 10
3
) = log 2 + log 10
3 = 0.30103 + 3 = 3.30103
log 2 = log (2 x 10°) = log 2 + log 10° = 0.30103 + 0 = 0.30103
log 0.000002 = log (2 x 108
) = log 2 + log 1Q6 = 0.30103 - 6 = -5.69897
Note that the characteristic is determined by the power to which 10 is raised
(when the number is in standard scientific notation), and the mantissa is determined by the log of the lefthand factor (when the number is in scientific notation). It is these properties that make it so easy to find the logarithm of a number
using a log table. Here is how you can do it.
1. Write the number (N) in standard scientific notation.
2. Look up the mantissa in the log table. It is the log of the lefthand factor
in scientific notation, which is a number between 1 and 10. The mantissa will lie between 0 and 1.
3. The exponent of 10 (the righthand factor) is the characteristic of the log.
4. Add the mantissa and the characteristic to obtain log N.
PROBLEM:
Find the log of 203.