problem solving logarithmic equation and reaching an equivalence ok so i've had a problem trying to simplify the  $\ln\left[ \sqrt{1+\frac{u^2}{a^2}} + \frac{u}{a} \right]$ and this is supposed to be equal to :  $\ln [ \sqrt{a^2+u^2} + u ]$
how is this posible ?? i've tried to solve this for more than 2 hours and couldn't get to this equivalence. any suggestions ?

$\ln x$ is injective so, if $\ln x=\ln y\implies x=y$. So, the following should hold true:\begin{align}
\sqrt{1+\frac{u^2}{a^2}}+\frac{u}{a}=\sqrt{a^2+u^2}+u\end{align}
However, simply plugging in $u=1, a=2$ gives us
\begin{align*}
&\sqrt{1+\frac{1}{4}}+\frac{1}{2}=\sqrt{4+1}+1\\
&\implies \sqrt{\frac{5}{4}}+\frac{1}{2}=\sqrt{5}+1\\
&\implies \sqrt{\frac{5}{4}}=\sqrt{5}-\frac{1}{2}\\
&\implies \frac{\sqrt{5}}{2}=\frac{2\sqrt{5}-1}{2}\\
&\implies \sqrt{5}=2\sqrt{5}-1
\end{align*}
which is false.
Hence they are not the same.
