Solve 0.02*(2/3)^x=0.1%

$$0.02 \times ( \frac { 2 } { 3 } ) ^ { x } = 0.1 \%$$

Solve for x

$$x=-\log_{\frac{2}{3}}\left(20\right)\approx 7.388384879$$

Steps Using Definition of Logarithm

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

$$0.02\times \left(\frac{2}{3}\right)^{x}=0.001$$

Multiply both sides by 50.

$$\left(\frac{2}{3}\right)^{x}=0.05$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{2}{3}\right)^{x})=\log(0.05)$$

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

$$x\log(\frac{2}{3})=\log(0.05)$$

Divide both sides by \log(\frac{2}{3}).

$$x=\frac{\log(0.05)}{\log(\frac{2}{3})}$$

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

$$x=\log_{\frac{2}{3}}\left(0.05\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(\frac{2}{3})}-\log_{\frac{2}{3}}\left(20\right)$$

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