11. Solve for x in the equation [ 25^x-5^2 x-1=500 ( (smiks) ) ]

To solve for \( x \) in the equation \( 25^x - 5^{2x-1} = 500 \), we can start by expressing both terms with the same base.<br /><br />First, note that \( 25 = 5^2 \). Therefore, we can rewrite \( 25^x \) as \( (5^2)^x = 5^{2x} \).<br /><br />Now, the equation becomes:<br />\[ 5^{2x} - 5^{2x-1} = 500 \]<br /><br />Next, factor out \( 5^{2x-1} \) from both terms:<br />\[ 5^{2x-1}(5 - 1) = 500 \]<br /><br />Simplify the expression inside the parentheses:<br />\[ 5^{2x-1} \cdot 4 = 500 \]<br /><br />Divide both sides by 4:<br />\[ 5^{2x-1} = 125 \]<br /><br />Now, express 125 as a power of 5:<br />\[ 125 = 5^3 \]<br /><br />So, we have:<br />\[ 5^{2x-1} = 5^3 \]<br /><br />Since the bases are the same, we can equate the exponents:<br />\[ 2x - 1 = 3 \]<br /><br />Solve for \( x \):<br />\[ 2x = 4 \]<br />\[ x = 2 \]<br /><br />Therefore, the solution to the equation is \( x = 2 \).