# Logarithm

The (single-valued) logarithm $\log \colon \mathbb{C} \setminus (-\infty,0] \rightarrow \mathbb{C}$ defined by the formula $$\log(z) = \displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$ where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not intersect the set $(-\infty,0] \subset \mathbb{C}$.

Domain coloring of $\log$.

# Properties

Logarithm of a complex number

Derivative of the logarithm

Real and imaginary parts of log

Relationship between logarithm (multivalued) and logarithm

Logarithm of product is a sum of logarithms

Logarithm of a quotient is a difference of logarithms

Relationship between logarithm and positive integer exponents

Logarithm of 1

Logarithm diverges to negative infinity at 0 from right

Logarithm at minus 1

Logarithm at i

Logarithm at -i

Taylor series of $\log(1-z)$

Taylor series of $\log(1+z)$

Antiderivative of the logarithm

Z2F1(1,1;2,-z) equals log(1+z)

Exponential integral Ei series

Relationship between prime zeta, Möbius function, logarithm, and Riemann zeta

The prime number theorem

Series for log(z) for Re(z) greater than 1/2

Series for log(z) for absolute value of (z-1) less than 1

Series for log(z) for Re(z) greater than 0

Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1

Series for log(z+a) for positive a and Re(z) greater than -a

Relationship between logarithm and Mangoldt

Log e(z)=log(z)

Log 10(z)=log(z)/log(10)

Log 10(z)=log 10(e)log(z)

Log(z)=log(10)log 10(z)

Limit of log(x)/x^a=0

Limit of x^a log(x)=0

X/(1+x) less than log(1+x)

Log(1+x) less than x

X less than -log(1-x)

-log(1-x) less than x/(1-x)

Abs(log(1-x)) less than 3x/2

Log(x) less than or equal to x-1

Log(x) less than or equal to n(x^(1/n)-1)

Abs(log(1+z)) less than or equal to -log(1-abs(z))

Log(1+z) as continued fraction

Log((1+z)/(1-z)) as continued fraction

# References

- 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.1.1$ - 1964: Milton Abramowitz and Irene A. Stegun:
*Handbook of mathematical functions*... (previous) ... (next): $4.1.48$

**Logarithm and friends**