Solve 10000^30=1000^x

$$10000 ^ { 30 } = 1000 ^ { x }$$

Solve for x

$$x=40$$

Steps Using Definition of Logarithm

Calculate 10000 to the power of 30 and get 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

$$1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000=1000^{x}$$

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

$$1000^{x}=1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000$$

Take the logarithm of both sides of the equation.

$$\log(1000^{x})=\log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)$$

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

$$x\log(1000)=\log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)$$

Divide both sides by \log(1000).

$$x=\frac{\log(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)}{\log(1000)}$$

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

$$x=\log_{1000}\left(1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i\log(e)}{3}+40$$

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