Solve 2(3)^x-3+1=69

$$2 ( 3 ) ^ { x - 3 } + 1 = 69$$

Solve for x

$$x=\log_{3}\left(34\right)+3\approx 6.209831677$$

Steps Using Definition of Logarithm

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

$$2\times 3^{x-3}+1=69$$

Subtract 1 from both sides of the equation.

$$2\times 3^{x-3}=68$$

Divide both sides by 2.

$$3^{x-3}=34$$

Take the logarithm of both sides of the equation.

$$\log(3^{x-3})=\log(34)$$

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

$$\left(x-3\right)\log(3)=\log(34)$$

Divide both sides by \log(3).

$$x-3=\frac{\log(34)}{\log(3)}$$

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

$$x-3=\log_{3}\left(34\right)$$

Add 3 to both sides of the equation.

$$x=\log_{3}\left(34\right)-\left(-3\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}+\log_{3}\left(34\right)+3$$

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