Solve 0.81^x=0.05

$${ 0.81 }^{ x } =0.05$$

Solve for x

$$x=-\frac{\log_{0.9}\left(20\right)}{2}\approx 14.216579403$$

Steps Using Definition of Logarithm

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

$$0.81^{x}=0.05$$

Take the logarithm of both sides of the equation.

$$\log(0.81^{x})=\log(0.05)$$

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

$$x\log(0.81)=\log(0.05)$$

Divide both sides by \log(0.81).

$$x=\frac{\log(0.05)}{\log(0.81)}$$

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

$$x=\log_{0.81}\left(0.05\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(0.9)}-\frac{\log_{0.9}\left(20\right)}{2}$$

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