Solve (99/100)^10x=0.5

$$( \frac { 99 } { 100 } ) ^ { 10 x } = 0.5$$

Solve for x

$$x=-\frac{\log_{0.99}\left(2\right)}{10}\approx 6.896756394$$

Steps Using Definition of Logarithm

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

$$\left(\frac{99}{100}\right)^{10x}=0.5$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{99}{100}\right)^{10x})=\log(0.5)$$

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

$$10x\log(\frac{99}{100})=\log(0.5)$$

Divide both sides by \log(\frac{99}{100}).

$$10x=\frac{\log(0.5)}{\log(\frac{99}{100})}$$

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

$$10x=\log_{\frac{99}{100}}\left(0.5\right)$$

Divide both sides by 10.

$$x=-\frac{\frac{\ln(2)}{\ln(\frac{99}{100})}}{10}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{5\ln(0.99)}-\frac{\log_{0.99}\left(2\right)}{10}$$

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