<s>[INST] Understanding the solution to $4^{3x+6}+3^{2x+3}=4^{3x+4}+2\times3^{2x+4}$ For the last 5 hours I tried to solve the equation below (without luck).
$$4^{3x+6}+3^{2x+3}=4^{3x+4}+2\times3^{2x+4}$$
A friend of mine then told me that the solution is
$$\frac{\ln(16/3)}{\ln(3/8)}$$
Can anyone help me understanding why this is the solution? No matter how I turn it, I never get to this solution.
My last final solution was
$$\frac{\ln(135)-\ln(16)}{\ln(64)-\ln(9)}$$
I'm looking forward to your answers, as I'm quite desperate at the moment.
 [/INST] $$4^{3x+6}+3^{2x+3}=4^{3x+4}+2 \times 3^{2x+4}$$
$$\Longleftrightarrow 16(4^{3x+4})+3^{2x+3}=4^{3x+4}+6(3^{2x+3}) $$
$$\Longleftrightarrow 15(4^{3x+4})=5(3^{2x+3}) $$
$$\Longleftrightarrow 3(4^{3x+4})=3^{2x+3} $$
$$\Longleftrightarrow \ln(3)+(3x+4)\ln(4)=(2x+3)\ln(3)$$
$$\Longleftrightarrow x(3\ln(4)-2\ln(3))=2\ln(3)-4\ln(4)$$
$$\Longleftrightarrow x=\frac{2\ln(3)-8\ln(2)}{6\ln(2)-2\ln(3)} = - \frac{\ln\left( \frac{16}{3}\right)}{\ln\left( \frac{8}{3}\right)}$$
so your friend is right !
</s>