It looks like you tried to write
e^4^x
which I would interpret as either
(so the exponent on <em>e</em> is 4<em>ˣ </em>)
or
(so the exponent on <em>e</em> is 4<em>x </em>)
as it's shown in the question body.
If you meant the first case,
![e^{4^x}-4=1](https://tex.z-dn.net/?f=e%5E%7B4%5Ex%7D-4%3D1)
![e^{4^x}=5](https://tex.z-dn.net/?f=e%5E%7B4%5Ex%7D%3D5)
![\ln\left(e^{4^x}\right)=\ln(5)](https://tex.z-dn.net/?f=%5Cln%5Cleft%28e%5E%7B4%5Ex%7D%5Cright%29%3D%5Cln%285%29)
![4^x\ln(e)=\ln(5)](https://tex.z-dn.net/?f=4%5Ex%5Cln%28e%29%3D%5Cln%285%29)
![4^x=\ln(5)](https://tex.z-dn.net/?f=4%5Ex%3D%5Cln%285%29)
![\log_4\left(4^x\right)=\log_4\left(\ln(5)\right)](https://tex.z-dn.net/?f=%5Clog_4%5Cleft%284%5Ex%5Cright%29%3D%5Clog_4%5Cleft%28%5Cln%285%29%5Cright%29)
![x\log_4(4)=\log_4\left(\ln(5)\right)](https://tex.z-dn.net/?f=x%5Clog_4%284%29%3D%5Clog_4%5Cleft%28%5Cln%285%29%5Cright%29)
![\boxed{x=\log_4\left(\ln(5)\right)}\approx0.3433](https://tex.z-dn.net/?f=%5Cboxed%7Bx%3D%5Clog_4%5Cleft%28%5Cln%285%29%5Cright%29%7D%5Capprox0.3433)
If you meant the second case,
![e^{4x}-4=1](https://tex.z-dn.net/?f=e%5E%7B4x%7D-4%3D1)
![e^{4x}=5](https://tex.z-dn.net/?f=e%5E%7B4x%7D%3D5)
![\ln\left(e^{4x}\right)=\ln(5)](https://tex.z-dn.net/?f=%5Cln%5Cleft%28e%5E%7B4x%7D%5Cright%29%3D%5Cln%285%29)
![4x\ln(e)=\ln(5)](https://tex.z-dn.net/?f=4x%5Cln%28e%29%3D%5Cln%285%29)
![4x=\ln(5)](https://tex.z-dn.net/?f=4x%3D%5Cln%285%29)
![\boxed{x=\dfrac{\ln(5)}4\approx0.4024}](https://tex.z-dn.net/?f=%5Cboxed%7Bx%3D%5Cdfrac%7B%5Cln%285%29%7D4%5Capprox0.4024%7D)