Find $x$ such that $ 2^{16^x} = 16^{2^x}.$
I am a bit confused, what will happen when we expand $16^{2^x}$, will we get $4^{2^{2^x}}$ or $4^{2^{x+1}}$ ?
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1$\begingroup$ $a^b a^c = a^{b+c}$ and $ a^{bc}={(a^b)^c}$.${(4^2)^{2^x}}={4^{2\times 2^x}}$ $\endgroup$– PNDasNov 10, 2020 at 12:00
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1$\begingroup$ $16^{2^x} = 16^{(2^x)} = (4^2)^{(2^x)} = 4^{2\cdot2^x} = 4^{(2^{x+1})}$ $\endgroup$– Jaap ScherphuisNov 10, 2020 at 12:03
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$\begingroup$ What they said. $a^{b^c}$ always means $a^{(b^c)}$ not $(a^b)^c$ because if you meant the latter, you'd simply write $a^{bc}$. $\endgroup$– PM 2RingNov 10, 2020 at 12:23
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3 Answers
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See that $2^{16^x} = 16^{2^x}$ is $2^{2^{4x}} = 2^{2^{x+2}}$
then $4x = x + 2$ and get $x = 2/3$.
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Alternative approach
$$16^{2^x} = [2^4]^{2^x} = 2^{4 \times 2^x}.$$
Since this is equal to $2^{16^x}$ you have
$$16^x = 4 \times 2^x \implies 8^x = 4 \implies x = (2/3).$$
answered Nov 10, 2020 at 12:20
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Take log on base 2, then $$16^x \log_2 2=2^x \log_2 16. \implies 16^x= 4. 2^x$ \implies 2^{4x}=2^{x+2} \implies 4x=x+2 \implies x=2/3.$$
