Solve 9000=3200(10775^x

$$9000=3200( { 10775 }^{ x }$$

Solve for x

$$x=\log_{10775}\left(\frac{45}{16}\right)\approx 0.11137055$$

Steps Using Definition of Logarithm

Divide both sides by 3200.

$$\frac{9000}{3200}=10775^{x}$$

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

$$\frac{45}{16}=10775^{x}$$

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

$$10775^{x}=\frac{45}{16}$$

Take the logarithm of both sides of the equation.

$$\log(10775^{x})=\log(\frac{45}{16})$$

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

$$x\log(10775)=\log(\frac{45}{16})$$

Divide both sides by \log(10775).

$$x=\frac{\log(\frac{45}{16})}{\log(10775)}$$

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

$$x=\log_{10775}\left(\frac{45}{16}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(10775)}+\log_{10775}\left(\frac{45}{16}\right)$$

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