Natural Logarithms Properties

Introduction to natural logarithm:The most important function-inverse pair in mathematics and science is the pair consisting of the natural logarithmfunction ln x and the exponential function ex. The key to understanding ex is ln x, so we introduce ln x first. The importance of logarithms came at first from the improvement they brought to arithmetic. The revolutionary properties of logarithms made possible the calculations of the great seventeenth-century advances in offshore navigation and celestial mechanics. Nowadays we do complicated arithmetic with calculators, but the properties of logarithms remain as important as ever.
natural Logarithm Function

The natural logarithm of a positive number x, written as ln x, is the value of an integral.Natural logarithm definition: The natural logarithmfunction

If x > 1, then ln x is the area under the curve y = 1/t from t = 1 to t = x. For 0 < x < 1, ln x gives the negative of the area under the curve from x to 1. The function is not defined for x = 0. We also have

Using x for everything would have us writing

with x meaning two different things. So, we change the variable of integration to t.
The graph of y = ln x and its relation to the function y = 1/x, x > 0. The graph of the logarithm rises above the x-axis as x moves from 1 to the right, and it falls below the axis as x moves from 1 to the left.

Properties of natural logarithms
The properties that made logarithms the single most important improvement in arithmetic before the advent of modern computers are listed below. The properties that made it possible to replace multiplication of positive numbers by addition and division of positive numbers by subtraction. They also made it possible to replace exponentiation by multiplication.
For any numbers a > 0 and x > 0
• The first property of Natural Logarithmsis Product rule , product of logarithm of x and y is equal to the sum of logarithm x and logarithm y: lnxy = ln x + ln y
• The second property of natural logarithmis quoitent rule, logarithm of division of x and y is equal to the difference of logarithm x and logarithm y:ln x/y = lnx – lny
• The third property of natural logarithm is Reciprocal rule:ln1/x = - ln x
• The forth property of natural logarithm is power rule: lnxn = n ln x.
Solving Natural Logarithms:  Let us understand the concept of solving natural logarithm with natural logarithm properties . Suppose we have to solve the natural logarithmwithout calculator ln e4. Now using the power property of natural logarithms, we have, 4 ln e. now simplifying using loge e = 1, we have the solution 4 . 1 = 4.

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