Solve 1500*1.4^x=2000*1.3^x

$$1500 \times { 1.4 }^{ x } = 2000 \cdot 1. { 3 }^{ x }$$

Solve for x

$$x=\log_{1.4}\left(\frac{4}{3}\right)\approx 0.854994978$$

Steps Using Definition of Logarithm

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

$$1500\times 1.4^{x}=2000$$

Divide both sides by 1500.

$$1.4^{x}=\frac{4}{3}$$

Take the logarithm of both sides of the equation.

$$\log(1.4^{x})=\log(\frac{4}{3})$$

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

$$x\log(1.4)=\log(\frac{4}{3})$$

Divide both sides by \log(1.4).

$$x=\frac{\log(\frac{4}{3})}{\log(1.4)}$$

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

$$x=\log_{1.4}\left(\frac{4}{3}\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.4)}+\log_{1.4}\left(\frac{4}{3}\right)$$

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