Solve 1000*3^x=80.000000

$$1000 \cdot 3 ^ { x } = 80.000000$$

Solve for x

$$x=\log_{3}\left(\frac{2}{25}\right)\approx -2.299017288$$

Steps Using Definition of Logarithm

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

$$1000\times 3^{x}=80$$

Divide both sides by 1000.

$$3^{x}=\frac{2}{25}$$

Take the logarithm of both sides of the equation.

$$\log(3^{x})=\log(\frac{2}{25})$$

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

$$x\log(3)=\log(\frac{2}{25})$$

Divide both sides by \log(3).

$$x=\frac{\log(\frac{2}{25})}{\log(3)}$$

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

$$x=\log_{3}\left(\frac{2}{25}\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(3)}+\log_{3}\left(\frac{2}{25}\right)$$

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