It looks like you tried to write
e^4^x
which I would interpret as either
[tex]e^{4^x}[/tex] (so the exponent on e is 4ˣ )
or
[tex]e^{4x}[/tex] (so the exponent on e is 4x )
as it's shown in the question body.
If you meant the first case,
[tex]e^{4^x}-4=1[/tex]
[tex]e^{4^x}=5[/tex]
[tex]\ln\left(e^{4^x}\right)=\ln(5)[/tex]
[tex]4^x\ln(e)=\ln(5)[/tex]
[tex]4^x=\ln(5)[/tex]
[tex]\log_4\left(4^x\right)=\log_4\left(\ln(5)\right)[/tex]
[tex]x\log_4(4)=\log_4\left(\ln(5)\right)[/tex]
[tex]\boxed{x=\log_4\left(\ln(5)\right)}\approx0.3433[/tex]
If you meant the second case,
[tex]e^{4x}-4=1[/tex]
[tex]e^{4x}=5[/tex]
[tex]\ln\left(e^{4x}\right)=\ln(5)[/tex]
[tex]4x\ln(e)=\ln(5)[/tex]
[tex]4x=\ln(5)[/tex]
[tex]\boxed{x=\dfrac{\ln(5)}4\approx0.4024}[/tex]
