Solve (1.25)^x=5000

$$( 1.25 ) ^ { x } = 5000$$

Solve for x

$$x=\log_{1.25}\left(5000\right)\approx 38.169120915$$

Steps Using Definition of Logarithm

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

$$1.25^{x}=5000$$

Take the logarithm of both sides of the equation.

$$\log(1.25^{x})=\log(5000)$$

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

$$x\log(1.25)=\log(5000)$$

Divide both sides by \log(1.25).

$$x=\frac{\log(5000)}{\log(1.25)}$$

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

$$x=\log_{1.25}\left(5000\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(1.25)}+\log_{1.25}\left(5000\right)$$

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