Find $x$ such that $ 2^{16^x} = 16^{2^x} $.

Question

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 $4^{2^{x+1}}$ ?

Answer 1

See that $2^{16^x} = 16^{2^x}$ is $2^{2^{4x}} = 2^{2^{x+2}}$

then $4x = x + 2$ and get $x = 2/3$.

Answer 2

Take log on base 2, then $$16^x \log_2 2=2^x \log_2 16. \implies 16^x= 4. 2^x$ \implies 2^{4x}=2^{x+2} \implies 4x=x+2 \implies x=2/3.$$

Answer 3

Alternative approach

$$16^{2^x} = [2^4]^{2^x} = 2^{4 \times 2^x}.$$

Since this is equal to $2^{16^x}$ you have

$$16^x = 4 \times 2^x \implies 8^x = 4 \implies x = (2/3).$$