Solve 1.5^x=10000

$$1.5 ^ { x } = 10000$$

Solve for x

$$x=4\log_{1.5}\left(10\right)\approx 22.715494349$$

Steps Using Definition of Logarithm

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

$$1.5^{x}=10000$$

Take the logarithm of both sides of the equation.

$$\log(1.5^{x})=\log(10000)$$

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

$$x\log(1.5)=\log(10000)$$

Divide both sides by \log(1.5).

$$x=\frac{\log(10000)}{\log(1.5)}$$

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

$$x=\log_{1.5}\left(10000\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.5)}+4\log_{1.5}\left(10\right)$$

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