Calculate: ((5)/(3))^{x-1}=((9)/(25))^{2x-7}

Expression: ${\left( \frac{ 5 }{ 3 } \right)}^{x-1}={\left( \frac{ 9 }{ 25 } \right)}^{2x-7}$

Write the expression in exponential form with the base of $\frac{ 5 }{ 3 }$

${\left( \frac{ 5 }{ 3 } \right)}^{x-1}={\left( \frac{ 5 }{ 3 } \right)}^{-4x+14}$

Since the bases are the same, set the exponents equal

$x-1=-4x+14$

Move the variable to the left-hand side and change its sign

$x-1+4x=14$

Move the constant to the right-hand side and change its sign

$x+4x=14+1$

Collect like terms

$5x=14+1$

$5x=15$
Divide both sides of the equation by $5$
$x=3$