Question

Giải phương trình :

                    \(\left(5^{x+2}\right)^{x+1}+\left(5^x\right)^{x+3}=\left(2^{x+1}\right)^{x+5}-6\left(2^{x+6}\right)^x\)

Answer 1

Phương trình đã cho tương đương với :

\(5^{\left(x+2\right)\left(x+1\right)}+5^{x\left(x+3\right)}=2^{\left(x+1\right)\left(x+5\right)}-6.2^{\left(x+6\right)x}\)

\(\Leftrightarrow5^{x^2+3x+2}+5^{x^2+3x}=2^{x^2+6x+5}-6.2^{x^2+6x}\)

\(\Leftrightarrow26.5^{x^2+3x}=26.2^{x^2+6x}\)

\(\Leftrightarrow5^{x^2+3x}=2^{x^2+6x}\)

\(\Leftrightarrow\left(x^2+3x\right)\log_25=x^2+6x\)

\(\Leftrightarrow x\left[\left(x+3\right)\log_25-\left(x+6\right)\right]=0\)

\(\Leftrightarrow\left[\begin{array}{nghiempt}x=0\\x=\frac{6-3\log_25}{\log_25-1}=\log_{\frac{5}{2}}\frac{64}{125}\end{array}\right.\)