A core exponential or logarithmic function has some important
algebraic properties with regard to addition and multiplication which
are commonly treated whenever the logarithm is introduced.
These key properties played a very significant role as a tool for simplifying calculations since the early 17th century.
Exponential Form |
Logarithmic Form |
|
Sum:Product |
$\exp_{base}(a+x) = \exp_{base}(a) \cdot \exp_{base}(x)$ ${base}^{a+x} = {base}^a \cdot {base}^x$ |
$\log_{base}(b\cdot y) = \log_{base}(b) + \log_{base}(y)$ [$b , y \gt 0$] |
Difference:Quotient |
$\exp_{base}(x-a) =\frac {\exp_{base}(x)}{\exp_{base}(a)}$ ${base}^{x-a} =\frac {{base}^x}{{base}^a}$ |
$\log_{base}(\frac yb) = \log_{base}(y) - \log_{base}(b)$ [$b , y \gt 0$] |
Product:Power |
$\exp_{base}(x\cdot p) =(\exp_{base}(x))^p$ ${base}^{x \cdot p} =({base}^x)^p$ |
$\log_{base}(y^p) = p \cdot \log_{base}(y) $ [$ y \gt 0$] |
Linear Function Composition | Exponential Form | Logarithmic Form |
Sum:Product | $\exp_{base} (l_{+a}(x)) =\exp_{base}(a+x)$ $ = \exp_{base}(a) \cdot \exp_{base}(x) =l_{\cdot \exp_{base}(a)}( \exp_{base}(x)) $ | $\log_{base}(l_{\cdot b}(y)) =
\log_{base}( b\cdot y)$ $ = \log_{base}(b) + \log_{base}(y) = l_{+log_{base}(b)}(\log_{base}(y))$ [$b , y \gt 0$] |
Difference:Quotient | $\exp_{base}
(l_{-a}(x))= \exp_{base}(x-a) $ $=\frac {\exp_{base}(x)}{\exp_{base}(a)} = l_{\cdot 1/ \exp_{base}(a)}( \exp_{base}(x)) $ | $\log_{base}(l_{\cdot 1/b}(y))
=\log_{base}(\frac yb)$ $ = log_{base}(y) - log_{base}(b)= l_{-log_{base}(b)}(\log_{base}(y))$ [$b , y \gt 0$] |
Product:Power | $\exp_{base}(l_{\cdot p} (x)) =\exp_{base}(x\cdot p) $ $ = (\exp_{base}(x))^p =\exp_{\exp_{base}(x)}(p)$ | $\log_{base}(\exp_y(p)) = \log_{base}(y^p)$ $ = p \cdot \log_{base}(y) =l_{\cdot p}( \log_{base}(y)) $ [$ y \gt 0$] |