Solve 1=(50,000)(0.95)^x

$$1 = ( 50,000 ) ( 0.95 ) ^ { x }$$

Solve for x

$$x=-\log_{0.95}\left(50000\right)\approx 210.939430068$$

Steps Using Definition of Logarithm

Divide both sides by 50000.

$$\frac{1}{50000}=0.95^{x}$$

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

$$0.95^{x}=\frac{1}{50000}$$

Take the logarithm of both sides of the equation.

$$\log(0.95^{x})=\log(\frac{1}{50000})$$

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

$$x\log(0.95)=\log(\frac{1}{50000})$$

Divide both sides by \log(0.95).

$$x=\frac{\log(\frac{1}{50000})}{\log(0.95)}$$

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

$$x=\log_{0.95}\left(\frac{1}{50000}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.95)}-\log_{0.95}\left(50000\right)$$

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