Solve 9000=3200(1.0775^x

$$9000=3200( { 1.0775 }^{ x }$$

Solve for x

$$x=\log_{1.0775}\left(2.8125\right)\approx 13.85349256$$

Steps Using Definition of Logarithm

Divide both sides by 3200.

$$\frac{9000}{3200}=1.0775^{x}$$

Reduce the fraction \frac{9000}{3200} to lowest terms by extracting and canceling out 200.

$$\frac{45}{16}=1.0775^{x}$$

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

$$1.0775^{x}=\frac{45}{16}$$

Take the logarithm of both sides of the equation.

$$\log(1.0775^{x})=\log(\frac{45}{16})$$

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

$$x\log(1.0775)=\log(\frac{45}{16})$$

Divide both sides by \log(1.0775).

$$x=\frac{\log(\frac{45}{16})}{\log(1.0775)}$$

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

$$x=\log_{1.0775}\left(\frac{45}{16}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.0775)}+\log_{1.0775}\left(2.8125\right)$$

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