Solve 8000=4500(1.024)^x

$$8000 = 4500 ( 1.024 ) ^ { x }$$

Solve for x

$$x=2\log_{1.024}\left(\frac{4}{3}\right)\approx 24.260050984$$

Steps Using Definition of Logarithm

Divide both sides by 4500.

$$\frac{8000}{4500}=1.024^{x}$$

Reduce the fraction \frac{8000}{4500} to lowest terms by extracting and canceling out 500.

$$\frac{16}{9}=1.024^{x}$$

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

$$1.024^{x}=\frac{16}{9}$$

Take the logarithm of both sides of the equation.

$$\log(1.024^{x})=\log(\frac{16}{9})$$

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

$$x\log(1.024)=\log(\frac{16}{9})$$

Divide both sides by \log(1.024).

$$x=\frac{\log(\frac{16}{9})}{\log(1.024)}$$

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

$$x=\log_{1.024}\left(\frac{16}{9}\right)$$