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# Algebra

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

Jun 5, 2024

#1
+1790
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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
+129829
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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 !!!

Jun 5, 2024