Solve 4^5*2^-4=2^k

$$4 ^ { 5 } \times 2 ^ { - 4 } = 2 ^ { k }$$

Solve for k

$$k=6$$

Steps Using Definition of Logarithm

Calculate 4 to the power of 5 and get 1024.

$$1024\times 2^{-4}=2^{k}$$

Calculate 2 to the power of -4 and get \frac{1}{16}.

$$1024\times \frac{1}{16}=2^{k}$$

Multiply 1024 and \frac{1}{16} to get 64.

$$64=2^{k}$$

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

$$2^{k}=64$$

Take the logarithm of both sides of the equation.

$$\log(2^{k})=\log(64)$$

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

$$k\log(2)=\log(64)$$

Divide both sides by \log(2).

$$k=\frac{\log(64)}{\log(2)}$$

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

$$k=\log_{2}\left(64\right)$$

Solve for k (complex solution)

$$k=\frac{2\pi n_{1}i}{\ln(2)}+6$$

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