The natural logarithm of 1 is 0. This is because the natural logarithm, denoted as ln(x), is the inverse function of the exponential function e^x.
To understand why this is the case, we can think about the definition of the natural logarithm: ln(x) answers the question, ‘To what power must e (approximately 2.71828) be raised, to produce x?’ When we set x to 1, we are looking for the power to which e must be raised to yield 1.
Mathematically, we can express this as:
e^y = 1
Here, y is the natural log we are trying to find. The only exponent that satisfies this equation is 0 because any number (except for 0) raised to the power of 0 equals 1. Therefore:
e^0 = 1
Thus, we conclude that:
ln(1) = 0
This property is important in mathematics, especially in calculus and growth modeling.