3. Find all solutions of the equation below. \( 5^{5-2 x}=5^{6 x-3} \)

To solve the equation \( 5^{5-2 x}=5^{6 x-3} \), we can take advantage of the property that if the bases are the same, then the exponents must be equal. Therefore, we set the exponents equal to each other: \[ 5 - 2x = 6x - 3 \] Next, we will combine like terms. Start by adding \( 2x \) to both sides: \[ 5 = 8x - 3 \] Now, add \( 3 \) to both sides: \[ 8 = 8x \] To isolate \( x \), divide both sides by \( 8 \): \[ x = 1 \] Thus, the solution to the equation is \( x = 1 \). To verify, substitute \( x \) back into the original equation: For \( x = 1 \): \[ 5^{5-2(1)} = 5^{6(1)-3} \] \[ 5^{5-2} = 5^{6-3} \] \[ 5^3 = 5^3 \] This confirms our solution is correct! Therefore, the only solution is \( x = 1 \).