j) \( 7^{x-2}=2^{x-2} \)

To solve the equation \( 7^{x-2} = 2^{x-2} \), start by taking the logarithm of both sides. You can use any logarithm, but using natural log (ln) or base 10 log is common. This gives you: \[ \ln(7^{x-2}) = \ln(2^{x-2}) \] Using the power rule of logarithms, you can rewrite this as: \[ (x-2) \ln(7) = (x-2) \ln(2) \] Now, if \( x - 2 = 0 \), you can find a trivial solution \( x = 2 \). If \( x - 2 \neq 0\), you can divide both sides by \( x - 2 \) (but beware of division by zero!). This leads to: \[ \ln(7) = \ln(2) \] Which is not true since \( \ln(7) \) is greater than \( \ln(2) \). Therefore, the only solution to the original equation is: \[ x = 2 \]