Introduction Exponential and Logarithmic Series

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In this chapter, we shall study two series known as the Exponential series and Logarithmic series. In our discussion, we shall make use of mathematical tools like formula for sum of an infinite G.P., combinatorial coefficients, the inequality 2^{n - 1} n! for n N etc.In this chapter, we shall study two series known as the Exponential series and Logarithmic series. In our discussion, we shall make use of mathematical tools like formula for sum of an infinite G.P., combinatorial coefficients, the inequality 2^{n - 1} n! for n N etc.

We know that 2^{n - 1} n! for n N.

Putting n = 2, 3, 4 ....., we get

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Adding vertically, we get

We know that log₂ 8 is the number to which 2 must be raised to get 8. Therefore, log₂ 8 = 3. In general, if a^x = y, (a > 0), then we say that log_a y = x. If e^x = y, then we say that the natural logarithm of y is x and we write log y = x. In other words, if the base of a logarithm is not mentioned, then it is understood that the base is e. In fact, we cannot think of logarithm of a number without a base.