Solve (1.075)^8x=1600

$$( 1.075 ) ^ { 8 x } = 1600$$

Solve for x

$$x = \frac{\log_{1.075} {(40)}}{4} \approx 12.751817301$$

Steps Using Definition of Logarithm

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

$$1.075^{8x}=1600$$

Take the logarithm of both sides of the equation.

$$\log(1.075^{8x})=\log(1600)$$

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

$$8x\log(1.075)=\log(1600)$$

Divide both sides by \log(1.075).

$$8x=\frac{\log(1600)}{\log(1.075)}$$

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

$$8x=\log_{1.075}\left(1600\right)$$

Divide both sides by 8.

$$x=\frac{2\ln(40)}{8\ln(\frac{43}{40})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{4\ln(1.075)}+\frac{\log_{1.075}\left(40\right)}{4}$$

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