Solve {3}^(frac{x{2}-1)}=10

$${ 3 }^{ ( \frac{ x }{ 2 } -1) } =10$$

Solve for x

$$x=2\left(\log_{3}\left(10\right)+1\right)\approx 6.191806549$$

Steps Using Definition of Logarithm

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

$$3^{\frac{1}{2}x-1}=10$$

Take the logarithm of both sides of the equation.

$$\log(3^{\frac{1}{2}x-1})=\log(10)$$

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

$$\left(\frac{1}{2}x-1\right)\log(3)=\log(10)$$

Divide both sides by \log(3).

$$\frac{1}{2}x-1=\frac{\log(10)}{\log(3)}$$

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

$$\frac{1}{2}x-1=\log_{3}\left(10\right)$$

Add 1 to both sides of the equation.

$$\frac{1}{2}x=\log_{3}\left(10\right)-\left(-1\right)$$

Multiply both sides by 2.

$$x=\frac{\log_{3}\left(10\right)+1}{\frac{1}{2}}$$

Solve for x (complex solution)

$$x=\frac{4\pi n_{1}i}{\ln(3)}+2\log_{3}\left(30\right)$$

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