Solve 55=4(2)^frac{x{4}}

$$55 = 4 ( 2 ) ^ { \frac { x } { 4 } }$$

Solve for x

$$x=4\left(\log_{2}\left(55\right)-2\right)\approx 15.125438854$$

Steps Using Definition of Logarithm

Divide both sides by 4.

$$\frac{55}{4}=2^{\frac{x}{4}}$$

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

$$2^{\frac{x}{4}}=\frac{55}{4}$$

Use the rules of exponents and logarithms to solve the equation.

$$2^{\frac{1}{4}x}=\frac{55}{4}$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{4}x})=\log(\frac{55}{4})$$

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

$$\frac{1}{4}x\log(2)=\log(\frac{55}{4})$$

Divide both sides by \log(2).

$$\frac{1}{4}x=\frac{\log(\frac{55}{4})}{\log(2)}$$

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

$$\frac{1}{4}x=\log_{2}\left(\frac{55}{4}\right)$$

Multiply both sides by 4.

$$x=\frac{\log_{2}\left(55\right)-2}{\frac{1}{4}}$$

Solve for x (complex solution)

$$x=\frac{8\pi n_{1}i}{\ln(2)}+4\log_{2}\left(55\right)-8$$

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