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
$exp_b(0) = 1$
$exp_b( r+s) = exp_b(r) \cdot exp_b(s)$ $log_b(M \cdot N) = log_b(M) + log_b(N)$
$exp_b( r-s) = \frac {exp_b(r)} {exp_b(s)}$
$exp_b( -s) =  \frac 1 {exp_b(s)}$

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