(1) $ \ln\left(x\right) = y $ :: $ e^y = x $
(2) $ \ln\left(xz\right) = \ln\left(x\right) + \ln\left(z\right) $
(3) $ \ln\left(x^a\right) = a + \left(\ln\left(x\right)\right) $
(1) $ \left(e^x\right)\left(e^y\right) = e^\left(x+y\right) $
(2) $ \left(e^x\right)^a =e^\left(a+x\right) $
(3) $ e^\left(\ln\left(x\right)\right) = x $
(1) $ \ln\left(e^x\right) = x $
(2) $ \frac{d}{dt}\left(e^{kx}\right) = e^{kx} $
(3) $ \frac{d}{dt}\left(\ln(x)\right) = \frac{1}{x}\frac{dx}{dt} $
(1) $ \ln(Y) - \ln(L) = \left(\frac{\alpha}{1-\alpha}\right) $
$ \left(\ln(K) - \ln(Y)\right) + \ln(E) $
(2) $ e^\left(\ln(Y) - \ln(L)\right) =
e^\left(\left(\frac{\alpha}{1-\alpha}\right) \left(\ln(K) - \ln(Y)\right) + \ln(E)\right) $
(3) $ Y ÷ L = \left(K ÷ Y\right)^\left(\frac{\alpha}{1-\alpha}\right)E^{1-\alpha} $
$ \ln\left(x\right) = y $ :: $ e^y = x $
$ \ln\left(xz\right) = \ln\left(x\right) + \ln\left(z\right) $
$ \ln\left(x^a\right) = a\left(\ln\left(x\right)\right) $
$ \left(e^x\right)\left(e^y\right) = e^\left(x+y\right) $
$ \left(e^x\right)^a =e^\left(ax\right) $
$ e^\left(\ln\left(x\right)\right) = x $
$ \ln\left(e^x\right) = x $
$ \frac{d}{dt}\left(e^{kx}\right) = ke^{kx} $
$ \frac{d}{dt}\left(\ln(x)\right) = \frac{1}{x}\frac{dx}{dt} $
$ \ln(Y) - \ln(L) = \left(\frac{\alpha}{1-\alpha}\right) $
$ \left(\ln(K) - \ln(Y)\right) + \ln(E) $
$ e^\left(\ln(Y) - \ln(L)\right) =
e^\left(\left(\frac{\alpha}{1-\alpha}\right) \left(\ln(K) - \ln(Y)\right) + \ln(E)\right) $
$ Y ÷ L = \left(K ÷ Y\right)^\left(\frac{\alpha}{1-\alpha}\right)E $
$ x_t = x_0\left(e^{gt}\right) $ vs.
$ x_t = x_0\left(1+g'\right)^t $
$ \ln{x_t} = \ln\left(x_0\left(e^{gt}\right)\right) $ vs.
$ \ln{x_t} = \ln\left(x_0\left(1+g'\right)^t\right) $
$ \ln{x_t} = \ln{x_0} + \ln\left(e^{gt}\right) $ vs.
$ \ln{x_t} = \ln{x_0} + \ln\left(\left(1+g'\right)^t\right) $
$ \ln{x_t} = \ln{x_0} + gt\ln\left(e\right) $ vs.
$ \ln{x_t} = \ln{x_0} +t\ln\left(1+g'\right) $
$ \ln{x_t} = \ln{x_0} + gt $ vs.
$ \ln{x_t} = \ln{x_0} + gt $ with $ g = \ln\left(1+g'\right) $
$ x_t = x_0\left(e^{gt}\right) $ ⇒ $ \frac{d}{dt}\left(\ln{x_t}\right) =
\frac{d}{dt}\left(\ln{x_0}\right) + \frac{d}{dt}\left(gt\right) = g $
$ x_t = x_0\left(e^{gt}\right) $ ⇒
$ \frac{dx_t}{dt} = x_0\left(ge^{gt}\right) = g\left(x_0e^{gt}\right) = gx_t $