Solve 2^632^616^3/256^4={4}^x

$${ 2 }^{ 6 } \frac{ { 32 }^{ 6 } { 16 }^{ 3 } }{ { 256 }^{ 4 } } = { 4 }^{ x }$$

Solve for x

$$x=8$$

Steps Using Definition of Logarithm

Calculate 2 to the power of 6 and get 64.

$$64\times \frac{32^{6}\times 16^{3}}{256^{4}}=4^{x}$$

Calculate 32 to the power of 6 and get 1073741824.

$$64\times \frac{1073741824\times 16^{3}}{256^{4}}=4^{x}$$

Calculate 16 to the power of 3 and get 4096.

$$64\times \frac{1073741824\times 4096}{256^{4}}=4^{x}$$

Multiply 1073741824 and 4096 to get 4398046511104.

$$64\times \frac{4398046511104}{256^{4}}=4^{x}$$

Calculate 256 to the power of 4 and get 4294967296.

$$64\times \frac{4398046511104}{4294967296}=4^{x}$$

Divide 4398046511104 by 4294967296 to get 1024.

$$64\times 1024=4^{x}$$

Multiply 64 and 1024 to get 65536.

$$65536=4^{x}$$

Swap sides so that all variable terms are on the left hand side.

$$4^{x}=65536$$

Take the logarithm of both sides of the equation.

$$\log(4^{x})=\log(65536)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$x\log(4)=\log(65536)$$

Divide both sides by \log(4).

$$x=\frac{\log(65536)}{\log(4)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$x=\log_{4}\left(65536\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(2)}+8$$

$$n_{1}\in \mathrm{Z}$$