Solve 20=50(.92)^x

$$20 = 50 ( .92 ) ^ { x }$$

Solve for x

$$x=\log_{0.92}\left(0.4\right)\approx 10.989122704$$

Steps Using Definition of Logarithm

Divide both sides by 50.

$$\frac{20}{50}=0.92^{x}$$

Reduce the fraction \frac{20}{50} to lowest terms by extracting and canceling out 10.

$$\frac{2}{5}=0.92^{x}$$

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

$$0.92^{x}=\frac{2}{5}$$

Take the logarithm of both sides of the equation.

$$\log(0.92^{x})=\log(\frac{2}{5})$$

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

$$x\log(0.92)=\log(\frac{2}{5})$$

Divide both sides by \log(0.92).

$$x=\frac{\log(\frac{2}{5})}{\log(0.92)}$$

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

$$x=\log_{0.92}\left(\frac{2}{5}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.92)}+\log_{0.92}\left(0.4\right)$$

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