Solve 1.80434=1.0225^4x

$$1.80434= { 1.0225 }^{ 4x }$$

Solve for x

$$x = \frac{\log_{1.0225} {(1.80434)}}{4} \approx 6.631221596$$

Steps Using Definition of Logarithm

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

$$1.0225^{4x}=1.80434$$

Take the logarithm of both sides of the equation.

$$\log(1.0225^{4x})=\log(1.80434)$$

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

$$4x\log(1.0225)=\log(1.80434)$$

Divide both sides by \log(1.0225).

$$4x=\frac{\log(1.80434)}{\log(1.0225)}$$

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

$$4x=\log_{1.0225}\left(1.80434\right)$$

Divide both sides by 4.

$$x=\frac{\ln(\frac{90217}{50000})}{4\ln(\frac{409}{400})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{2\ln(1.0225)}-\frac{\log_{1.0225}\left(\frac{50000}{90217}\right)}{4}$$

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