Solve 0.3^x-2=1/27

$$0.3 ^ { x - 2 } = \frac { 1 } { 27 }$$

Solve for x

$$x=-\log_{0.3}\left(300\right)\approx 4.737467868$$

Steps Using Definition of Logarithm

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

$$0.3^{x-2}=\frac{1}{27}$$

Take the logarithm of both sides of the equation.

$$\log(0.3^{x-2})=\log(\frac{1}{27})$$

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

$$\left(x-2\right)\log(0.3)=\log(\frac{1}{27})$$

Divide both sides by \log(0.3).

$$x-2=\frac{\log(\frac{1}{27})}{\log(0.3)}$$

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

$$x-2=\log_{0.3}\left(\frac{1}{27}\right)$$

Add 2 to both sides of the equation.

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

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.3)}-\log_{0.3}\left(300\right)$$

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