247. При каком натуральном значении х выполняется равенство: a) \frac{3^{2x}}{4^x} = 2,25; B) \frac{3^{x+1} \cdot 9^x}{27} = 3; б) \frac{2^{3x-1} \cdot 16}{4^x} = 64; r) г) \frac{5^x \cdot 25^x}{125^x} = 5^{x-1}?

а)

\[\frac{3^{2x}}{4^x} = 2.25\]

\[\frac{9^x}{4^x} = \frac{9}{4}\]

\[(\frac{9}{4})^x = \frac{9}{4}\]

\[x = 1\]

б)

\[\frac{2^{3x-1} \cdot 16}{4^x} = 64\]

\[\frac{2^{3x-1} \cdot 2^4}{2^{2x}} = 2^6\]

\[2^{3x-1+4-2x} = 2^6\]

\[2^{x+3} = 2^6\]

\[x+3 = 6\]

\[x = 3\]

в)

\[\frac{3^{x+1} \cdot 9^x}{27} = 3\]

\[\frac{3^{x+1} \cdot 3^{2x}}{3^3} = 3^1\]

\[3^{x+1+2x-3} = 3^1\]

\[3^{3x-2} = 3^1\]

\[3x-2 = 1\]

\[3x = 3\]

\[x = 1\]

г)

\[\frac{5^x \cdot 25^x}{125^x} = 5^{x-1}\]

\[\frac{5^x \cdot 5^{2x}}{5^{3x}} = 5^{x-1}\]

\[\frac{5^{3x}}{5^{3x}} = 5^{x-1}\]

\[1 = 5^{x-1}\]

\[5^0 = 5^{x-1}\]

\[0 = x-1\]

\[x = 1\]