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1. Domain:

$\begin{array}{|rcll|} \hline 5^{x+1}-20 &> & x 0 \quad &| \quad +20 \\ 5^{x+1} &>& 20 \quad &| \quad 5^{x+1}=5^x\cdot 5^1 \\ 5^x\cdot 5^1 &>& 20 \\ 5^x\cdot 5 &>& 4\cdot 5 \quad &| \quad : 5\\ 5^x &=& 4 \quad &| \quad \ln()\\ \ln(5^x) &>& \ln(4) \\ x\cdot \ln(5) &>& \ln(4) \quad &| \quad : \ln(5)\\ x &>& \frac{ \ln(4) } { \ln(5) } \\ x &>& \frac{ 1.38629436112 } { 1.60943791243 } \\ \mathbf{ x } & \mathbf{>} & \mathbf{ 0.86135311615 } \\ \hline \end{array}$

2. Solution:

$\begin{array}{|rcll|} \hline \log_5{5^{x+1}-20} &=& x \quad &| \quad 5^{()} \\ 5^{x+1}-20 &=& 5^x \quad &| \quad 5^{x+1}=5^x\cdot 5^1 \\ 5^x\cdot 5^1 -20 &=& 5^x \quad &| \quad +20 \\ 5^x\cdot 5 &=& 5^x +20 \quad &| \quad -5^x \\ 5^x\cdot 5 -5^x &=& 20 \\ 5^x\cdot (5-1) &=& 20 \\ 5^x\cdot 4 &=& 20 \quad &| \quad : 4 \\ 5^x &=& \frac{20}{4} \\ 5^x &=& 5 \\ 5^x &=& 5^1 \\ \mathbf{ x } & \mathbf{=} & \mathbf{1} \\ \hline \end{array}$

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