Solve 2500(1,048)^x=4,000

$$2500 ( 1,048 ) ^ { x } = 4,000$$

Solve for x

$$x=\log_{1048}\left(\frac{8}{5}\right)\approx 0.067581313$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$2500\times 1048^{x}=4000$$

Divide both sides by 2500.

$$1048^{x}=\frac{8}{5}$$

Take the logarithm of both sides of the equation.

$$\log(1048^{x})=\log(\frac{8}{5})$$

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

$$x\log(1048)=\log(\frac{8}{5})$$

Divide both sides by \log(1048).

$$x=\frac{\log(\frac{8}{5})}{\log(1048)}$$

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

$$x=\log_{1048}\left(\frac{8}{5}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(1048)}+\log_{1048}\left(\frac{8}{5}\right)$$

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