Solve $LaTeX: \displaystyle 3^{x + 2}=5^{x}$ .

Taking the natural logarithm of both sides gives $LaTeX: \displaystyle (x + 2)\ln(3)=x\ln(5)$ . Distributing gives $LaTeX: \displaystyle x \ln{\left(3 \right)} + 2 \ln{\left(3 \right)} = x \ln{\left(5 \right)}$ . Moving all the $LaTeX: \displaystyle x$ terms to the left hand side and all of the constants to the right side gives: $LaTeX: \displaystyle - x \ln{\left(5 \right)} + x \ln{\left(3 \right)} = - 2 \ln{\left(3 \right)}$ . Factoring out the $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x \left(- \ln{\left(5 \right)} + \ln{\left(3 \right)}\right)=- 2 \ln{\left(3 \right)}$ . Isolating $LaTeX: \displaystyle x$ yeilds $LaTeX: \displaystyle x = \frac{- \ln{\left(3 \right)}}{- \ln{\left(5 \right)} + \ln{\left(3 \right)}}$ .