Solve 2(3^x-1)=4

$$2 ( 3 ^ { x - 1 } ) = 4$$

Solve for x

$$x=\log_{3}\left(2\right)+1\approx 1.630929754$$

Steps Using Definition of Logarithm

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

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

Divide both sides by 2.

$$3^{x-1}=2$$

Take the logarithm of both sides of the equation.

$$\log(3^{x-1})=\log(2)$$

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

$$\left(x-1\right)\log(3)=\log(2)$$

Divide both sides by \log(3).

$$x-1=\frac{\log(2)}{\log(3)}$$

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

$$x-1=\log_{3}\left(2\right)$$

Add 1 to both sides of the equation.

$$x=\log_{3}\left(2\right)-\left(-1\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}+\log_{3}\left(6\right)$$

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