Solve 2*16^2*32^7=2^x

$$2 \cdot 16 ^ { 2 } \cdot 32 ^ { 7 } = 2 ^ { x }$$

Solve for x

$$x=44$$

Steps Using Definition of Logarithm

Calculate 16 to the power of 2 and get 256.

$$2\times 256\times 32^{7}=2^{x}$$

Multiply 2 and 256 to get 512.

$$512\times 32^{7}=2^{x}$$

Calculate 32 to the power of 7 and get 34359738368.

$$512\times 34359738368=2^{x}$$

Multiply 512 and 34359738368 to get 17592186044416.

$$17592186044416=2^{x}$$

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

$$2^{x}=17592186044416$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(17592186044416)$$

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

$$x\log(2)=\log(17592186044416)$$

Divide both sides by \log(2).

$$x=\frac{\log(17592186044416)}{\log(2)}$$

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

$$x=\log_{2}\left(17592186044416\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(2)}+44$$

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