euler's constant

diff --git a/spec/calculus.md b/spec/calculus.md
index 4c3331860356d8310bfdf56c7fb075fbc790b0dc..1f3d404a908eb59d373868cb5710f97b5ac51fe3 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -130,3 +130,25 @@ If $f(x)={u(x) \over v(x)}$, then $f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)
 
 If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
 
 
 If $y={u(x) \over v(x)}$, then derivative ${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$
 

+## Logarithms
+
+$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
+
+Wikipedia:
+
+> the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$
+
+### Logarithmic identities  
+$\log_b (xy)=\log_b x + \log_b y$  
+$\log_b x^n = n \log_b x$  
+$\log_b y^{x^n} = x^n \log_b y$
+
+### $e$ as a logarithm
+
+$$\log_e e = 1$$
+$$\ln x = \log_e x$$
+
+### Differentiating logarithms
+$${d \over dx} \log_b x = {1 \over x \ln b}$$
+
+