Solve 2.9^x=0.2

$$2.9 ^ { x } = 0.2$$

Solve for x

$$x=-\log_{2.9}\left(5\right)\approx -1.511619876$$

Steps Using Definition of Logarithm

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

$$2.9^{x}=0.2$$

Take the logarithm of both sides of the equation.

$$\log(2.9^{x})=\log(0.2)$$

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

$$x\log(2.9)=\log(0.2)$$

Divide both sides by \log(2.9).

$$x=\frac{\log(0.2)}{\log(2.9)}$$

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

$$x=\log_{2.9}\left(0.2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(2.9)}-\log_{2.9}\left(5\right)$$

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