<s>[INST] Find all real $x$ for which $\log_{10}(2^{x-1}+3^{x+1})=2x-\log_2{3^x}$ My attempt:
$$\log_{10}(2^{x-1}+3^{x+1})=x\cdot(2-\log_2{3})$$
I am stuck after this. If the terms inside LHS were a product, this problem would have been a piece of cake. The domain of $x$ is all real here so I can't limit the range of $x$ for a hit and trial solution. 
Any starting steps please?
 [/INST] Hint: power each side by 2. 
$$A = 2^{x-1} + 3^{x+1} = 2^{2x-xlog_2(3)}$$
$$= (2^{2-log_2(3)})^x = \left(\frac{4}{3}\right)^x$$
$$\Rightarrow 2^{x-1}3^x+3^{2x+1} = 2^{2x}$$
Suppose $X = 2^x$, $Y=3^x$:
$$\frac{1}{2}XY+ 3Y^2 = X^2$$
$$ X^2 - 3Y^2 - \frac{1}{2}XY = 0$$
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