Solve 180434=10225^4x

$$180434= { 10225 }^{ 4x }$$

Solve for x

$$x=\frac{\log_{10225}\left(180434\right)}{4}\approx 0.327728163$$

Steps Using Definition of Logarithm

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

$$10225^{4x}=180434$$

Take the logarithm of both sides of the equation.

$$\log(10225^{4x})=\log(180434)$$

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

$$4x\log(10225)=\log(180434)$$

Divide both sides by \log(10225).

$$4x=\frac{\log(180434)}{\log(10225)}$$

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

$$4x=\log_{10225}\left(180434\right)$$

Divide both sides by 4.

$$x=\frac{\log_{10225}\left(180434\right)}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(10225)}+\frac{\log_{10225}\left(180434\right)}{4}$$

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