Solve 0.1=0.9985^x

$$0.1= { 0.9985 }^{ x }$$

Solve for x

$$x=-\log_{0.9985}\left(10\right)\approx 1533.905148077$$

Steps Using Definition of Logarithm

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

$$0.9985^{x}=0.1$$

Take the logarithm of both sides of the equation.

$$\log(0.9985^{x})=\log(0.1)$$

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

$$x\log(0.9985)=\log(0.1)$$

Divide both sides by \log(0.9985).

$$x=\frac{\log(0.1)}{\log(0.9985)}$$

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

$$x=\log_{0.9985}\left(0.1\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(0.9985)}-\log_{0.9985}\left(10\right)$$

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