Solve 108^x=2

$$108 ^ { x } = 2$$

Solve for x

$$x=\log_{108}\left(2\right)\approx 0.148040955$$

Steps Using Definition of Logarithm

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

$$108^{x}=2$$

Take the logarithm of both sides of the equation.

$$\log(108^{x})=\log(2)$$

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

$$x\log(108)=\log(2)$$

Divide both sides by \log(108).

$$x=\frac{\log(2)}{\log(108)}$$

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

$$x=\log_{108}\left(2\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(108)}+\log_{108}\left(2\right)$$

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