LOGARITHM -- NEED OF EXTENSION OF EXPONENTS, DEFINITION, FUNCTION, LINKS TO FURTHER STUDY
Before you proceed , Look at the need of Logarithm in Exponents,
explained at the end, by means of an example,if you have not already done so. Also knowledge of Exponents is a prerequisite here.
Look at a question in exponents: what is x if 7x = 343. The same question can be asked here by using a symbol.The question is " what is log7 343 ? " If you understand this algebraic symbol, which you will, a little later, you will understand the question as find x such that 7x = 343. More over while solving the question, you will write 7x = 343 = 73 ⇒ x = 3. Since 343 is 73, you could easily answer it. If I give you find x such that 7x = 100, what will you do? With the symbolism you learn here, you will answer it as x = log7 100. It is not only the symbolism for finding the exponents, but a useful body of knowledge, you will learn here.
Definition of Logarithm :
If a(≠1) and n are any positive real numbers
and for some real x, ax = n,
then x is said to be the logarithm of n to the base a.
It is written as logan = x.
For real values of a, x, n and a≠1, a > 0, n > 0 ax = n ⇒ logan = x
34 = 81 ⇒ log3 81 = 4
92 = 81 ⇒ log9 81 = 2
From these examples we can conclude that the logs of the same
number (here 81) to different bases (here 3 and 9) are different (4 and 2). i.e.
The logs of the same number to different bases are different.
Some more examples:
641⁄3 = 4 ⇒ log64 4 = 1⁄3
5-2 = 1⁄25 ⇒ log5 (1⁄25) = -2
You may note the following points.
Base of the log (a in definition) is positive and not equal to 1.
Here logs are defined only for positive real numbers (n in definition).
The value of the log (x in definition) can be negative.
ax = n is called exponential form.
logan = x is called logarithmic form.
Both in exponential form and logarithmic form, the base is same.
The exponent in exponential form is the value of the log
(to the same base to which the exponent is raised) in logarithmic form.
Also there exists a unique x which satisfies the equation
ax = n. So logan is unique.
Look at the following two sets of examples 1st set: 23 = 8 ⇒ log2 8 = 3 24 = 16 ⇒ log2 16 = 4 25 = 32 ⇒ log2 32 = 5 Here as the number (8, 16, 32) increases,the log of the number (3, 4, 5) increases. Note that the base (2) is more than 1. What is seen to be true here in this example, is infact true in general. The general statement is
If a > 1, n1 > n2 ⇒ logan1 > logan2 i.e. when the base is more than 1, the logarithmic function is an increasing function.
2nd set: (1⁄2)5 = 1⁄32 ⇒ log(1⁄2) (1⁄32) = 5 (1⁄2)4 = 1⁄16 ⇒ log(1⁄2) (1⁄16) = 4 (1⁄2)3 = 1⁄8 ⇒ log(1⁄2) (1⁄8) = 3 Here as the number (1⁄32, 1⁄16, 1⁄8) increases,the log of the number (5, 4, 3) decreases. Note that the base (1⁄2) is less than 1. What is seen to be true here in this example, is infact true in general. The general statement is
If 0 < a < 1, n1 > n2 ⇒ logan1 < logan2 i.e. when the base is less than 1 (and positive), the logarithmic function is a decreasing function.
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