Solve 3^frac{x{2}+1}=12.2

$$3 ^ { \frac { x } { 2 } + 1 } = 12.2$$

Solve for x

$$x=2\log_{3}\left(12.2\right)-2\approx 2.553810252$$

Steps Using Definition of Logarithm

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

$$3^{\frac{1}{2}x+1}=12.2$$

Take the logarithm of both sides of the equation.

$$\log(3^{\frac{1}{2}x+1})=\log(12.2)$$

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

$$\left(\frac{1}{2}x+1\right)\log(3)=\log(12.2)$$

Divide both sides by \log(3).

$$\frac{1}{2}x+1=\frac{\log(12.2)}{\log(3)}$$

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

$$\frac{1}{2}x+1=\log_{3}\left(12.2\right)$$

Subtract 1 from both sides of the equation.

$$\frac{1}{2}x=\frac{\ln(\frac{61}{5})}{\ln(3)}-1$$

Multiply both sides by 2.

$$x=\frac{\frac{\ln(\frac{61}{5})}{\ln(3)}-1}{\frac{1}{2}}$$

Solve for x (complex solution)

$$x=\frac{i\times 4\pi n_{1}}{\ln(3)}+2\log_{3}\left(12.2\right)-2$$

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