Mathematics Logarithms
In mathematics logarithms were developed for making complicated calculations simple.
For example, if a right circular cylinder has radius r = 0.375 meters and height h = 0.2321 meters, then its volume is given by: V = A = πr
^{2}h = 3.146 × (0.375)
^{2} × 0.2321. Use for logarithm tables makes such calculations quite easy. However, even calculators have functions like multiplication; power etc. still, logarithmic and exponential equations and functions are very common in mathematics.
Definition:
If a
^{x} = M (M > 0, a > 0, a ≠ 1), then x (i.e., index of the power) is called the logarithm of the number
M to the base
a and is written as
x = log_{a} M.
Hence, if a
^{x} = M then x = log
_{a} M;
conversely, if x = log
_{a} M then a
^{x} = M.
If ‘
a’ is a positive real number (except 1),
n is any real number and
a^{n} = b, then
n is called the
logarithm of b to the base a.
It is written as log
_{a} b (read as log of b to the base a).
Thus,
a^{n} = b ⇔ log_{a} b = n.
a
^{n} is called the exponential form and log
_{a} b = n is called the logarithmic form.
For example:
● 3
^{2} = 9 ⇔ log
_{3} 9 = 2
● 5
^{4} = 625 ⇔ log
_{5} 625 = 4
● 7
^{0} = 1 ⇔ log
_{7} 1 = 0
● 2
^{3} =
^{1}/
_{8} ⇔ log
_{2} (
^{1}/
_{8}) = 3
● 10
^{2} = 0.01 ⇔ log
_{10} 0.01 = 2
● 2
^{6} = 64 ⇔ log
_{2} 64 = 6
● 3
^{ 4} = 1/3
^{4} = 1/81 ⇔ log
_{3} 1/81 = 4
● 10
^{2} = 1/100 = 0.01 ⇔ log
_{10} 0.01 = 2
Notes on basic Logarithm Facts:
1. Since a > 0 (a ≠ 1), a
^{n} > 0 for any rational n. Hence logarithm is defined only positive real numbers.
From the definition it is clear that the logarithm of a number has no meaning if the base is not mentioned.
2. The above examples shows that the logarithm of a (positive) real number may be negative, zero or positive.
3. Logarithmic values of a given number are different for different bases.
4. Logarithms to the base a 10 are called
common logarithms. Also,
logarithm tables assume base 10. If no base is given, the base is assumed to be 10.
For example: log 21 means log
_{10} 21.
5. Logarithm to the base ‘
e’ (where e = 2.7183 approx.) is called
natural logarithm, and is usually written as
ln. Thus ln x means log
_{e} x.
6. If a
^{x} =  M (a > 0, M > 0), then the value of x will be imaginary i.e., logarithmic value of a negative number is imaginary.
7. Logarithm of 1 to any finite nonzero base is zero.
Proof: We know, a
^{0} = 1 (a ≠ 0). Therefore, from the definition, we have, log
_{a} 1 = 0.
8. Logarithm of a positive number to the same base is always 1.
Proof: Since a
^{1} = a. Therefore, log
_{a} a = 1.
Note:
From 7 and 8 we say that,
log_{a} 1 = 0 and
log_{a} a = 1 for any positive real ‘a’ except 1.
9. If x = log
_{a} M then a
^{log a M} = a
Proof:
x = log
_{a} M. Therefore, a
^{x} = M or, a
^{log}_{a}^{M} = M [Since, x = log
_{a} M].
● Mathematics Logarithm
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Logarithm Rules or Log Rules
Solved Problems on Logarithm
Common Logarithm and Natural Logarithm
Antilogarithm
Logarithms
11 and 12 Grade Math
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