Solve 4^7*16*8^-4/32^-2=2^x+13

$$\frac{ { 4 }^{ 7 } \times 16 \times { 8 }^{ -4 } }{ { 32 }^{ -2 } } = { 2 }^{ x+13 }$$

Solve for x

$$x=3$$

Steps Using Definition of Logarithm

Calculate 4 to the power of 7 and get 16384.

$$\frac{16384\times 16\times 8^{-4}}{32^{-2}}=2^{x+13}$$

Multiply 16384 and 16 to get 262144.

$$\frac{262144\times 8^{-4}}{32^{-2}}=2^{x+13}$$

Calculate 8 to the power of -4 and get \frac{1}{4096}.

$$\frac{262144\times \frac{1}{4096}}{32^{-2}}=2^{x+13}$$

Multiply 262144 and \frac{1}{4096} to get 64.

$$\frac{64}{32^{-2}}=2^{x+13}$$

Calculate 32 to the power of -2 and get \frac{1}{1024}.

$$\frac{64}{\frac{1}{1024}}=2^{x+13}$$

Divide 64 by \frac{1}{1024} by multiplying 64 by the reciprocal of \frac{1}{1024}.

$$64\times 1024=2^{x+13}$$

Multiply 64 and 1024 to get 65536.

$$65536=2^{x+13}$$

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

$$2^{x+13}=65536$$

Take the logarithm of both sides of the equation.

$$\log(2^{x+13})=\log(65536)$$

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

$$\left(x+13\right)\log(2)=\log(65536)$$

Divide both sides by \log(2).

$$x+13=\frac{\log(65536)}{\log(2)}$$

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

$$x+13=\log_{2}\left(65536\right)$$

Subtract 13 from both sides of the equation.

$$x=16-13$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(2)}+3$$

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