Solve (0.9)^x=0.5

$$( 0.9 ) ^ { x } = 0.5$$

Solve for x

$$x=-\log_{0.9}\left(2\right)\approx 6.578813479$$

Steps Using Definition of Logarithm

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

$$0.9^{x}=0.5$$

Take the logarithm of both sides of the equation.

$$\log(0.9^{x})=\log(0.5)$$

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

$$x\log(0.9)=\log(0.5)$$

Divide both sides by \log(0.9).

$$x=\frac{\log(0.5)}{\log(0.9)}$$

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

$$x=\log_{0.9}\left(0.5\right)$$

Solve for x (complex solution)

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

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