The Medical College - Page 11
REFERENCE DATA FOR MODEL TEST
Logarithms and Exponents
Logarithms
The logarithm of any number is the exponent of
the power to which 10 must be raised to pro-
duce the number. The logarithm X of the number N
to the base 10 is the exponent of the power
to which 10 must be raised to give N (for
example, log
10 N = X). Logarithms
consist of two
parts. First, there is the "characteristic,"
which is determined by the position of the first
signifi-
cant figure of the number in relation to the
decimal point. If we count leftwards from the deci-
mal point as positive and rightwards as
negative, the characteristic is equal to the count
ending
at the right of the first significant figure.
Thus, the characteristic of the logarithm of 2340 is
3,
and of 0.00234 is –3. Second, there is the
"mantissa." It is always positive, is found in loga-
rithm tables, and depends only on the sequence
of significant figures. Thus, the mantissa for the
two numbers is the same, namely 0.3692. The
logarithm of a number is the sum of the charac-
teristic and the mantissa. Thus, log 2340 =
3.3692 while log 0.00234 = –3 + 0.3692 =
–2.6308.
The logarithms of the whole integers 1 to 10
are given below.
log 1.0 = 0.000
log 4.0 = 0.602
log 7.0 = 0.845
log 10.0 = 1.000
log 2.0 = 0.301
log 5.0 = 0.699
log 8.0 = 0.903
log 3.0 = 0.477
log 6.0 = 0.778
log 9.0 = 0.954
Useful Rules in Handling
Logarithms
1. The logarithm of a product is equal to the
sum of the logarithm of the factors:
log ab = log a + log b
(Check this out by solving for log 6, using log
2 + log 3.)
2. The logarithm of a fraction is equal to the
logarithm of the numerator minus the
logarithm of the denominator:
log a
b
= log a – log b
Example: log 10
2
= log 10 – log 2 = log 5
How about log 2.5? The answer from the log
tables is 0.398.
3. The logarithm of the reciprocal of a number
is the negative logarithm of the number:
log 1
a = log 1 – log a
Since log 1 = 0, then
log 1
a = – log a
Equally,
log 1
2
= – log 2 = –0.301
4. The logarithm of a number raised to a power
is the logarithm of the number
multiplied by the power:
log a
b = b log
a
log 2
2 = 0.603
Exponents
It is convenient to express large numbers as
10
x
, where x represents the number of
places that the decimal must be moved to place
it after the first significant figure. This
also represents 10 · 10 for x times. For
example, 1,000,000 may be expressed as 1
10
6;
3663 as 3.663
10
3; and so on. To
multiply, the exponents are added, but coefficients
are multiplied. To divide, the exponents are
subtracted but coefficients are divided.
Multiplying: (1
10
x) ·
(1
10
y) = 1
10
x+y
(4
10
2) ·
(2
10
3) = 8
10
5
Dividing: (1
10
x) ÷
(1
10
y) = 1
10
x–y
(4
10
2) ÷
(2
10
3) = 2
10
–1
Numbers less than 1 are 10
–x. For example,
0.000001 is 1
10
–6.
Multiplying: (1
10
–x) ·
(1
10
–y) =
1
10
–(x+y)
(4
10
–2) ·
(2
10
–3) =
8
10
–5
A large number multiplied by a small number:
(4
10
–2)(2
10
3) = 8
10
1
(Logarithms and Exponents are reproduced
through the courtesy of Dr. Richard B.
Brandt, Dept. of Biochemistry, MCV, VCU,
Richmond, Virginia, 23298).
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