Solve 14,400=4000(1,025)^x

$$14,400 = 4000 ( 1,025 ) ^ { x }$$

Solve for x

$$x=\log_{1025}\left(\frac{18}{5}\right)\approx 0.184773671$$

Steps Using Definition of Logarithm

Divide both sides by 4000.

$$\frac{14400}{4000}=1025^{x}$$

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

$$\frac{18}{5}=1025^{x}$$

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

$$1025^{x}=\frac{18}{5}$$

Take the logarithm of both sides of the equation.

$$\log(1025^{x})=\log(\frac{18}{5})$$

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

$$x\log(1025)=\log(\frac{18}{5})$$

Divide both sides by \log(1025).

$$x=\frac{\log(\frac{18}{5})}{\log(1025)}$$

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

$$x=\log_{1025}\left(\frac{18}{5}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(1025)}+\log_{1025}\left(\frac{18}{5}\right)$$

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