Today is our topic of discussion Concept of Logarithm .
Concept of Logarithm
Logarithm originated from two Greek words Logos and arithmas. Logos means discussion and arithmas means number. So logarithm means discussion about numbers.
Definition: If ax = b, where a > 0 and a1 then z is called the logarithm of b to the base a where x = log b
Hence, if ax=b, then x = logxab
On the other hand, if x = log,(ab), then ax = b
In this case the number b is the antilogarithm of r with respect to base a and we write, b = antilog(ax)
In many cases the bases of log and anti log is not written.
Note: The approximate value of loga can be determined using scientific calculator. (Detailed explanation is given in the mathematics book of
According to the definition, we get,
log_2(64) = 6 as 2^6 = 64 and log8(64) = 2 as 8² = 64
Therefore, logarithm of same number can be different based on different bases. Different logarithm values of same number can be determined by taking any number as base which is positive but not 1. Any positive number can be regarded as a base of logarithm. Logarithm values of zero or any negative number can not be determined.
Note: If a > 0 ,a ≠1 and b ne0 then logarithm of b with a unique base a can be denoted by log_a(b)
Therefore,
1) log_a(b) = x if and only if a ^ x = b From (a) we see that,
2) log_a(a^x) = x
3) a ^ log_a(b) = b
Formulas of Logarithm
Since proofs have been already given in the mathematics book of class 9-10, here only the formulas are shown.
1. log_a(a) = 1 and log_a(1) = 0
2. log_a(MN) = log_a(M) + log_a(N)
3. loga (~) = log₁M – logN N
4. loga (MN) Nlog, M
5. log, M = log, Mx log b [Formula for changing the base]
Exponential, Logarithmic and Absolute Value Functions
In the first chapter, we learnt about function, in this chapter we shall know about Exponential, Logarithmic and Absolute Value Functions.
Exponential Function
Observe the following three tables of corresponding values of x and y:
Table 1
Table 2
Table 3
In Table 1, for different values of z we get values of y of equal differences complying with the function y = 2x This is an equation of straight line.
In Table 2, the same happens complying with the function y = x²
In Table 3, the same is true for y = 2^x
Exponential function f(x) = a^x is true for all real number x. Here, a > 0 and a ≠1
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