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avatar+476 

Find x if \log_2 (\log_3 x) = 2 \log_4 (x).

 Jun 5, 2024
 #1
avatar+759 
+1

First, let's apply some log rules to simplify our equation. We get

\(\log _3\left(x\right)=x\) from this. 

 

We know \(\mathrm{If}\:f\left(x\right)=g\left(x\right),\:\mathrm{then}\:a^{f\left(x\right)}=a^{g\left(x\right)}\)

From this, we have \(3^{\log _3\left(x\right)}=3^x\)

Which is the same as \(x=3^x\)

 

There are NO SOLUTIONS to this equation. 

 

Thanks! :)

 Jun 5, 2024
 #2
avatar+129399 
+1

Thanks, NotThat Smart !!!!

 

\(\log_2 (\log_3 x) = 2 \log_4 (x) \)

 

To elaborate a little, we can use the change of base theorem to write

 

log ( logx / log 3 )            2 log (x)

______________    =    ________                { log 4  = log 2^2  = 2log 2 }

  log 2                             log 4

 

log ( log x / log 3 )           2log (x)

______________  =     _______

      log 2                        2log 2

 

log ( logx / log 3 ) =      log (x)

 

This implies that

 

logx / log 3  = x

 

log x = xlog 3

 

log x  = log 3^x

 

x = 3^x

 

Note here that the graphs of both functions never intersect

https://www.desmos.com/calculator/82kerdz2ip

 

So, like NotThatSmart, I don't find any solutions, either !!!

 

 

cool cool cool

 Jun 5, 2024

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