The solution set is square Solve the equation 9^3x-7=81^x+3

To solve the equation \(9^{3x-7} = 81^{x+3}\), we need to express both sides of the equation with the same base. Notice that \(9\) and \(81\) can both be written as powers of \(3\):<br /><br />\[ 9 = 3^2 \]<br />\[ 81 = 3^4 \]<br /><br />Now, rewrite the equation using these expressions:<br /><br />\[ (3^2)^{3x-7} = (3^4)^{x+3} \]<br /><br />Simplify the exponents:<br /><br />\[ 3^{2(3x-7)} = 3^{4(x+3)} \]<br /><br />This gives us:<br /><br />\[ 3^{6x-14} = 3^{4x+12} \]<br /><br />Since the bases are the same, we can set the exponents equal to each other:<br /><br />\[ 6x - 14 = 4x + 12 \]<br /><br />Solve for \(x\):<br /><br />\[ 6x - 4x = 12 + 14 \]<br />\[ 2x = 26 \]<br />\[ x = 13 \]<br /><br />Thus, the solution set is \(\{ 13 \}\).