Solve 1000000=4186.58*left(1.0250right)^x

$$1000000=4186.58 \times { \left(1.0250 \right) }^{ x }$$

Solve for x

$$x=\log_{1.025}\left(\frac{50000000}{209329}\right)\approx 221.76151217$$

Steps Using Definition of Logarithm

Divide both sides by 4186.58.

$$\frac{1000000}{4186.58}=1.025^{x}$$

Expand \frac{1000000}{4186.58} by multiplying both numerator and the denominator by 100.

$$\frac{100000000}{418658}=1.025^{x}$$

Reduce the fraction \frac{100000000}{418658} to lowest terms by extracting and canceling out 2.

$$\frac{50000000}{209329}=1.025^{x}$$

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

$$1.025^{x}=\frac{50000000}{209329}$$

Take the logarithm of both sides of the equation.

$$\log(1.025^{x})=\log(\frac{50000000}{209329})$$

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

$$x\log(1.025)=\log(\frac{50000000}{209329})$$

Divide both sides by \log(1.025).

$$x=\frac{\log(\frac{50000000}{209329})}{\log(1.025)}$$

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

$$x=\log_{1.025}\left(\frac{50000000}{209329}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.025)}+\log_{1.025}\left(\frac{50000000}{209329}\right)$$

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