Solve 0.99^x=0.45

$${ 0.99 }^{ x } =0.45$$

Solve for x

$$x=\log_{0.99}\left(0.45\right)\approx 79.450847002$$

Steps Using Definition of Logarithm

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

$$0.99^{x}=0.45$$

Take the logarithm of both sides of the equation.

$$\log(0.99^{x})=\log(0.45)$$

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

$$x\log(0.99)=\log(0.45)$$

Divide both sides by \log(0.99).

$$x=\frac{\log(0.45)}{\log(0.99)}$$

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

$$x=\log_{0.99}\left(0.45\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.99)}+\log_{0.99}\left(0.45\right)$$

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