a) \(\left(3^x-5\right)^4=2^8\)
\(\Rightarrow\left(3^x-5\right)^4=\left(2^2\right)^4\)
\(\Rightarrow\left(3^x-5\right)^4=4^4\)
\(\Rightarrow3^x-5=4\)
\(\Rightarrow3^x=4+5\)
\(\Rightarrow3^x=9\)
\(\Rightarrow3^x=3^3\)
\(\Rightarrow x=3\)
Vậy \(x=3\).
b) \(3^{x+2}+3^x=10^2-10\)
\(\Rightarrow3^x\left(3^2+1\right)=10\left(10-1\right)\)
\(\Rightarrow3^x\cdot10=90\)
\(\Rightarrow3^x=\dfrac{90}{10}=9\)
\(\Rightarrow3^x=3^2\)
\(\Rightarrow x=2\)
Vậy \(x=2\).
c) \(3^{x+1}-3^x=4\cdot5-2\)
\(\Rightarrow3^x\left(3-1\right)=20-2\)
\(\Rightarrow3^x\cdot2=18\)
\(\Rightarrow3^x=\dfrac{18}{2}=9\)
\(\Rightarrow3^x=3^2\)
\(\Rightarrow x=2\)
Vậy \(x=2\).
d) \(5^{x+2}-5^x=10^3:2+10^2\)
\(\Rightarrow5^x\left(5^2-1\right)=1000:2+100\)
\(\Rightarrow5^x\cdot24=600\)
\(\Rightarrow5^x=\dfrac{600}{24}=25\)
\(\Rightarrow5^x=5^2\)
\(\Rightarrow x=2\)
Vậy \(x=2\).
e) \(4^x+4^{x-1}=84:7\)
\(\Rightarrow4^x:1+4^x:4=12\)
\(\Rightarrow4^x\cdot1+4^x\cdot\dfrac{1}{4}=12\)
\(\Rightarrow4^x\left(1+\dfrac{1}{4}\right)=12\)
\(\Rightarrow4^x\cdot\dfrac{5}{4}=12\)
\(\Rightarrow4^x=12:\dfrac{5}{4}=12\cdot\dfrac{4}{5}=9.6\)
\(\Rightarrow x\in\left\{\varnothing\right\}\)
Vậy \(x\in\left\{\varnothing\right\}\).
