Solve =14,400=4,000(1+25/100)^n

$$= 14,400 = 4,000 ( 1 + \frac { 25 } { 100 } ) ^ { n }$$

Solve for n

$$n=\log_{\frac{5}{4}}\left(\frac{18}{5}\right)\approx 5.740402705$$

Steps Using Definition of Logarithm

Divide both sides by 4000.

$$\frac{14400}{4000}=\left(1+\frac{25}{100}\right)^{n}$$

Reduce the fraction \frac{14400}{4000} to lowest terms by extracting and canceling out 800.

$$\frac{18}{5}=\left(1+\frac{25}{100}\right)^{n}$$

Reduce the fraction \frac{25}{100} to lowest terms by extracting and canceling out 25.

$$\frac{18}{5}=\left(1+\frac{1}{4}\right)^{n}$$

Add 1 and \frac{1}{4} to get \frac{5}{4}.

$$\frac{18}{5}=\left(\frac{5}{4}\right)^{n}$$

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

$$\left(\frac{5}{4}\right)^{n}=\frac{18}{5}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{5}{4}\right)^{n})=\log(\frac{18}{5})$$

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

$$n\log(\frac{5}{4})=\log(\frac{18}{5})$$

Divide both sides by \log(\frac{5}{4}).

$$n=\frac{\log(\frac{18}{5})}{\log(\frac{5}{4})}$$

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

$$n=\log_{\frac{5}{4}}\left(\frac{18}{5}\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(\frac{5}{4})}+\log_{\frac{5}{4}}\left(\frac{18}{5}\right)$$

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