Algebra of Exponential and Logarithmic functions:
The basic algebraic facts for exponential and logarithmic functions are discussed thoroughly in most second courses in algebra.

 Algebra for Exponential Functions Algebra for Logarithmic Functions $exp_b(log_b(y))=y$ $log_b(exp_b(x))=x$ $exp_b(0) = 1$ $log_b(1)=0$ $exp_b( r+s) = exp_b(r) \cdot exp_b(s)$ $log_b(M \cdot N) = log_b(M) + log_b(N)$ $exp_b( r-s) = \frac {exp_b(r)} {exp_b(s)}$ $exp_b( -s) = \frac 1 {exp_b(s)}$

$exp_{\frac 1 b}(x) = (\frac 1 b)^x = (b^{-1})^x = b^ {-x} = exp_b(-x)$
$log_b( 1/x) = -log_b(x)$
The basic fact for multiplicative inverses and exponents: $a^{-1} = \frac 1 a$.