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 = πr2
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.
= 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 = loga M
Hence, if ax
= M then x = loga
conversely, if x = loga
M then ax
’ is a positive real number (except 1), n
is any real number and an = b
, then n
is called the logarithm of b to the base a
It is written as loga
b (read as log of b to the base a).
an = b ⇔ loga b = n.
is called the exponential form and loga
b = n is called the logarithmic form.
= 9 ⇔ log3
9 = 2
= 625 ⇔ log5
625 = 4
= 1 ⇔ log7
1 = 0
) = -3
= 0.01 ⇔ log10
0.01 = -2
= 64 ⇔ log2
64 = 6
= 1/81 ⇔ log3
1/81 = -4
= 1/100 = 0.01 ⇔ log10
0.01 = -2
Notes on basic Logarithm Facts:
Since a > 0 (a ≠ 1), an
> 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.
The above examples shows that the logarithm of a (positive) real number may be negative, zero or positive.
Logarithmic values of a given number are different for different bases.
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 log10
Logarithm to the base ‘e
’ (where e = 2.7183 approx.) is called natural logarithm
, and is usually written as ln
. Thus ln x means loge
= - M (a > 0, M > 0), then the value of x will be imaginary i.e., logarithmic value of a negative number is imaginary.
Logarithm of 1 to any finite non-zero base is zero. Proof:
We know, a0
= 1 (a ≠ 0). Therefore, from the definition, we have, loga
1 = 0.
Logarithm of a positive number to the same base is always 1.
= a. Therefore, loga
a = 1.
From 7 and 8 we say that, loga 1 = 0
and loga a = 1
for any positive real ‘a’ except 1.
If x = loga
M then a log a M
x = loga
M. Therefore, ax
= M or, a logaM
= M [Since, x = loga
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