Solve 6*2^frac{x{4}}=222

$$6 \cdot 2 ^ { \frac { x } { 4 } } = 222$$

Solve for x

$$x=4\log_{2}\left(37\right)\approx 20.837813463$$

Steps Using Definition of Logarithm

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

$$6\times 2^{\frac{1}{4}x}=222$$

Divide both sides by 6.

$$2^{\frac{1}{4}x}=37$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{4}x})=\log(37)$$

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

$$\frac{1}{4}x\log(2)=\log(37)$$

Divide both sides by \log(2).

$$\frac{1}{4}x=\frac{\log(37)}{\log(2)}$$

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

$$\frac{1}{4}x=\log_{2}\left(37\right)$$

Multiply both sides by 4.

$$x=\frac{\log_{2}\left(37\right)}{\frac{1}{4}}$$

Solve for x (complex solution)

$$x=\frac{8\pi n_{1}i}{\ln(2)}+4\log_{2}\left(37\right)$$

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