ln1

ln1 denotes the natural logarithm of the number 1. The natural logarithm, written ln(x), is defined for positive real numbers x and is the inverse function of the exponential function exp(x) = e^x. By definition, ln(y) gives the unique real number x that satisfies e^x = y.

For y = 1, this means finding x such that e^x = 1. Since e^0 = 1, the solution is x = 0. Therefore, ln(1) = 0. This value is consistent with the general properties of logarithms: for any positive a, ln(a^k) = k ln(a) and ln(ab) = ln(a) + ln(b); applying these with a = b = 1 yields ln(1) = 0.

In addition to the exact value, ln(1) illustrates basic features of the logarithm: the natural logarithm is increasing, its domain is (0, ∞), and as x approaches 0 from the right, ln(x) tends to −∞ while as x grows, ln(x) grows without bound. The fact that ln(1) = 0 also underpins many calculus and algebraic identities involving logarithms and exponentials.