Logarithm

LOGARITHM

A Final Project for Algebra

Submitted by: Submitted to:

Nicolas, Deogracias A. Sir Kevin Tolentino

CSDP / P1

INTRODUCTION

In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. In simple cases the logarithm counts repeated multiplication. For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000 (1000 = 10 × 10 × 10 = 103); the multiplication is repeated three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers b and x where b is not equal to 1. The logarithm of x to base b, denoted logb (x), is the unique real number y such that by = x. For example, as 64 = 26, we have log2(64) = 6.

The idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the third power (or cube) of 2 is 8, because 8 is the product of three factors of 2:

23 = 2 x 2 x 2 = 8

It follows that the logarithm of 8 with respect to base 2 is 3, so log2 8 = 3.

Logarithm shows us of how many times does a number multiplied by itself but not written in exponential form, it can be the other way for writing exponents instead of using exponential form because logarithmic function is the inverse of exponential function.

ANALYTIC PROPERTIES OF LOGARITHM

Logarithmic Function

To justify the definition of logarithms, it is necessary to show that the equation b x = y

has a solution x and that this solution is unique, provided that y is positive and that b is positive and unequal to 1. A proof of that fact requires the intermediate value theorem from elementary...