Answer
$x=\text{ any real number}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
5^{2x-6}=25^{x-3}
,$ use the laws of exponents to express both sides in the same base. Then equate the exponents.
$\bf{\text{Solution Details:}}$
Using exponents, the given equation is equivalent to
\begin{array}{l}\require{cancel}
5^{2x-6}=(5^2)^{x-3}
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
5^{2x-6}=5^{2(x-3)}
.\end{array}
Since the bases are the same, then the exponents can be equated. That is,
\begin{array}{l}\require{cancel}
2x-6=2(x-3)
\\\\
2x-6=2(x)+2(-3)
\\\\
2x-6=2x-6
\\\\
2x-2x=-6+6
\\\\
0=0
\text{ (TRUE)}
.\end{array}
Since the equation above ended with a TRUE statement, the given equation is an identity. Hence, the values of $x$ that satisfy the given equation is $
x=\text{ any real number}
.$
