Solve 0.01^2x*1000=0.1

$${ 0.01 }^{ 2x } \times 1000 = 0.1$$

Solve for x

$$x=1$$

Steps Using Definition of Logarithm

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

$$1000\times 0.01^{2x}=0.1$$

Divide both sides by 1000.

$$0.01^{2x}=0.0001$$

Take the logarithm of both sides of the equation.

$$\log(0.01^{2x})=\log(0.0001)$$

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

$$2x\log(0.01)=\log(0.0001)$$

Divide both sides by \log(0.01).

$$2x=\frac{\log(0.0001)}{\log(0.01)}$$

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

$$2x=\log_{0.01}\left(0.0001\right)$$

Divide both sides by 2.

$$x=\frac{2}{2}$$

Solve for x (complex solution)

$$x=-\frac{i\pi n_{1}\log(e)}{2}+1$$

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