The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved Se mer

The natural logarithm can be defined in several equivalent ways.
Inverse of exponential
The most general definition is as the inverse function of Se mer

For $${\displaystyle \ln(x)}$$ where x > 1, the closer the value of x is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this:
Such techniques were … Se mer

The exponential function can be extended to a function which gives a complex number as e for any arbitrary complex number z; simply use … Se mer

The notations ln x and loge x both refer unambiguously to the natural logarithm of x, and log x without an explicit base may also refer to the … Se mer

Since the natural logarithm is undefined at 0, $${\displaystyle \ln(x)}$$ itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor … Se mer

While no simple continued fractions are available, several generalized continued fractions exist, including:
These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers … Se mer