<s>[INST] Logarithm / exponential equation, not sure what to make of this, (simple) Solve for $a:(2 \log_a x)(3 \log_{x^2} 4) = 3$
No idea how to approach this problem other than moving the 2 and the 3 into an exponent..
 [/INST] $(2 \log_a x)(3 \log_{x^2} 4) = 3\iff( \log_a x^2)( \log_{x^2} 4) = 1$. Then, because of the well known formula $\log_xy\log_y x=1$ one has $$\log_4 x^2\log_{x^2}4=1\Rightarrow\log_4 x^2=\frac{1}{\log_{x^2}4}=\log_a x^2\Rightarrow4^{x^2}=a^{x^2}\iff\left(\frac4a\right)^{x^2}=1$$
Thus (since $g(x)=b^x$ is injective) we have $a=4$.
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