Solve (1+0.12/12)^12n-1/frac{0.12{12}}=20417.42

$$\frac { ( 1 + \frac { 0.12 } { 12 } ) ^ { 12 n } - 1 } { \frac { 0.12 } { 12 } } = 20417.42$$

Solve for n

$$n = \frac{\log_{1.01} {(205.1742)}}{12} \approx 44.586954385$$

Steps Using Definition of Logarithm

Expand \frac{0.12}{12} by multiplying both numerator and the denominator by 100.

$$\frac{\left(1+\frac{12}{1200}\right)^{12n}-1}{\frac{0.12}{12}}=20417.42$$

Reduce the fraction \frac{12}{1200} to lowest terms by extracting and canceling out 12.

$$\frac{\left(1+\frac{1}{100}\right)^{12n}-1}{\frac{0.12}{12}}=20417.42$$

Add 1 and \frac{1}{100} to get \frac{101}{100}.

$$\frac{\left(\frac{101}{100}\right)^{12n}-1}{\frac{0.12}{12}}=20417.42$$

Expand \frac{0.12}{12} by multiplying both numerator and the denominator by 100.

$$\frac{\left(\frac{101}{100}\right)^{12n}-1}{\frac{12}{1200}}=20417.42$$

Reduce the fraction \frac{12}{1200} to lowest terms by extracting and canceling out 12.

$$\frac{\left(\frac{101}{100}\right)^{12n}-1}{\frac{1}{100}}=20417.42$$

Divide each term of \left(\frac{101}{100}\right)^{12n}-1 by \frac{1}{100} to get \frac{\left(\frac{101}{100}\right)^{12n}}{\frac{1}{100}}+\frac{-1}{\frac{1}{100}}.

$$\frac{\left(\frac{101}{100}\right)^{12n}}{\frac{1}{100}}+\frac{-1}{\frac{1}{100}}=20417.42$$

Divide -1 by \frac{1}{100} by multiplying -1 by the reciprocal of \frac{1}{100}.

$$\frac{\left(\frac{101}{100}\right)^{12n}}{\frac{1}{100}}-100=20417.42$$

Subtract 20417.42 from both sides.

$$\frac{\left(\frac{101}{100}\right)^{12n}}{\frac{1}{100}}-100-20417.42=0$$

Subtract 20417.42 from -100 to get -20517.42.

$$\frac{\left(\frac{101}{100}\right)^{12n}}{\frac{1}{100}}-20517.42=0$$

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

$$100\times \left(\frac{101}{100}\right)^{12n}-20517.42=0$$

Add 20517.42 to both sides of the equation.

$$100\times \left(\frac{101}{100}\right)^{12n}=20517.42$$

Divide both sides by 100.

$$\left(\frac{101}{100}\right)^{12n}=205.1742$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{101}{100}\right)^{12n})=\log(205.1742)$$

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

$$12n\log(\frac{101}{100})=\log(205.1742)$$

Divide both sides by \log(\frac{101}{100}).

$$12n=\frac{\log(205.1742)}{\log(\frac{101}{100})}$$

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

$$12n=\log_{\frac{101}{100}}\left(205.1742\right)$$

Divide both sides by 12.

$$n=\frac{\ln(\frac{1025871}{5000})}{12\ln(\frac{101}{100})}$$

Solve for n (complex solution)

$$n=\frac{i\pi n_{1}}{6\ln(1.01)}-\frac{\log_{1.01}\left(\frac{5000}{1025871}\right)}{12}$$

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