Solve 2500=1500(1+.06)^x

$$2500 = 1500 ( 1 + .06 ) ^ { x }$$

Solve for x

$$x=\log_{1.06}\left(\frac{5}{3}\right)\approx 8.766692911$$

Steps Using Definition of Logarithm

Divide both sides by 1500.

$$\frac{2500}{1500}=\left(1+0.06\right)^{x}$$

Reduce the fraction \frac{2500}{1500} to lowest terms by extracting and canceling out 500.

$$\frac{5}{3}=\left(1+0.06\right)^{x}$$

Add 1 and 0.06 to get 1.06.

$$\frac{5}{3}=1.06^{x}$$

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

$$1.06^{x}=\frac{5}{3}$$

Take the logarithm of both sides of the equation.

$$\log(1.06^{x})=\log(\frac{5}{3})$$

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

$$x\log(1.06)=\log(\frac{5}{3})$$

Divide both sides by \log(1.06).

$$x=\frac{\log(\frac{5}{3})}{\log(1.06)}$$

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

$$x=\log_{1.06}\left(\frac{5}{3}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.06)}+\log_{1.06}\left(\frac{5}{3}\right)$$

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