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avatar+1071 

Find $x$ if $\log_2 (\log_3 x) = \log_3 (\log_2 x).$

 Mar 27, 2024
 #1
avatar+394 
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\(\log_2(\log_3x)=\log_3(\log_2x) \rightarrow \text{ apply change of base to } \log_3 \text{ on both sides and change to } \log_2 \text{.}\)

\(\log_2(\frac{\log_2x}{\log_23})=\frac{\log_2(\log_2x)}{\log_23}\)

\(\log_2(\log_2x) - \log_2({\log_23})=\frac{\log_2(\log_2x)}{\log_23}\)

\(\log_2(\log_2x) \rightarrow z \; \log_23 \rightarrow y\)

\(z-\log_2y=\frac{z}{y}\)

\(z=\frac{y\log_2y}{y-1}\)

\(x=2^{2^{\frac{(\log_23)(\log_2(\log_23))}{\log_2{3}-1}}}\)

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 Mar 27, 2024
 #2
avatar+53 
0

I agree - 2^2 is 4, by the way

 Mar 27, 2024

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