Solve 23^x=3

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

Solve for x

$$x=-\log_{3}\left(2\right)+1\approx 0.369070246$$

Steps Using Definition of Logarithm

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

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

Divide both sides by 2.

$$3^{x}=\frac{3}{2}$$

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(3).

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

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

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

Solve for x (complex solution)

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

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