Solve 3*2^frac{x{5}}=150

$$3 \cdot 2 ^ { \frac { x } { 5 } } = 150$$

Solve for x

$$x=10\log_{2}\left(5\right)+5\approx 28.219280949$$

Steps Using Definition of Logarithm

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

$$3\times 2^{\frac{1}{5}x}=150$$

Divide both sides by 3.

$$2^{\frac{1}{5}x}=50$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{5}x})=\log(50)$$

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

$$\frac{1}{5}x\log(2)=\log(50)$$

Divide both sides by \log(2).

$$\frac{1}{5}x=\frac{\log(50)}{\log(2)}$$

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

$$\frac{1}{5}x=\log_{2}\left(50\right)$$

Multiply both sides by 5.

$$x=\frac{\log_{2}\left(50\right)}{\frac{1}{5}}$$

Solve for x (complex solution)

$$x=\frac{10\pi n_{1}i}{\ln(2)}+10\log_{2}\left(5\right)+5$$

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