Solve 1/16*2^frac{3{4}x}=4

$$\frac { 1 } { 16 } \cdot 2 ^ { \frac { 3 } { 4 } x } = 4$$

Solve for x

$$x=8$$

Steps Using Definition of Logarithm

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

$$\frac{1}{16}\times 2^{\frac{3}{4}x}=4$$

Multiply both sides by 16.

$$2^{\frac{3}{4}x}=64$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{3}{4}x})=\log(64)$$

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

$$\frac{3}{4}x\log(2)=\log(64)$$

Divide both sides by \log(2).

$$\frac{3}{4}x=\frac{\log(64)}{\log(2)}$$

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

$$\frac{3}{4}x=\log_{2}\left(64\right)$$

Divide both sides of the equation by \frac{3}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{6}{\frac{3}{4}}$$

Solve for x (complex solution)

$$x=\frac{8\pi n_{1}i}{3\ln(2)}+8$$

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