Solve 0.5=0.9^xdiv2

$$0.5 = { 0.9 }^{ x \div 2 }$$

Solve for x

$$x=-2\log_{0.9}\left(2\right)\approx 13.157626958$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$0.9^{\frac{x}{2}}=0.5$$

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

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(0.9).

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

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

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

Multiply both sides by 2.

$$x=-\frac{\frac{\ln(2)}{\ln(\frac{9}{10})}}{0.5}$$

Solve for x (complex solution)

$$x=\frac{i\times 4\pi n_{1}}{\ln(0.9)}-2\log_{0.9}\left(2\right)$$

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