Solve $LaTeX: \displaystyle 9^{x - 6}=4^{x}$ .

Taking the natural logarithm of both sides gives $LaTeX: \displaystyle (x - 6)\ln(9)=x\ln(4)$ . Distributing gives $LaTeX: \displaystyle 2 x \ln{\left(3 \right)} - 12 \ln{\left(3 \right)} = x \ln{\left(4 \right)}$ . Moving all the $LaTeX: \displaystyle x$ terms to the left hand side and all of the constants to the right side gives: $LaTeX: \displaystyle - x \ln{\left(4 \right)} + x \ln{\left(9 \right)} = 6 \ln{\left(9 \right)}$ . Factoring out the $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x \left(- \ln{\left(4 \right)} + \ln{\left(9 \right)}\right)=6 \ln{\left(9 \right)}$ . Isolating $LaTeX: \displaystyle x$ yeilds $LaTeX: \displaystyle x = \frac{- \ln{\left(9 \right)}}{- \ln{\left(4 \right)} + \ln{\left(9 \right)}}$ .