Solve 3000*1032^x=6000

$$3000 \times { 1032 }^{ x } =6000$$

Solve for x

$$x=\log_{1032}\left(2\right)\approx 0.099887853$$

Steps Using Definition of Logarithm

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

$$3000\times 1032^{x}=6000$$

Divide both sides by 3000.

$$1032^{x}=2$$

Take the logarithm of both sides of the equation.

$$\log(1032^{x})=\log(2)$$

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

$$x\log(1032)=\log(2)$$

Divide both sides by \log(1032).

$$x=\frac{\log(2)}{\log(1032)}$$

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

$$x=\log_{1032}\left(2\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(1032)}+\log_{1032}\left(2\right)$$

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