Solve (5.825)^(x-3)=120

$$( 5.825 ) ^ { ( x - 3 ) } = 120$$

Solve for x

$$x=-\log_{5.825}\left(\frac{1600}{37948011}\right)\approx 5.716833012$$

Steps Using Definition of Logarithm

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

$$5.825^{x-3}=120$$

Take the logarithm of both sides of the equation.

$$\log(5.825^{x-3})=\log(120)$$

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

$$\left(x-3\right)\log(5.825)=\log(120)$$

Divide both sides by \log(5.825).

$$x-3=\frac{\log(120)}{\log(5.825)}$$

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

$$x-3=\log_{5.825}\left(120\right)$$

Add 3 to both sides of the equation.

$$x=\frac{\ln(120)}{\ln(\frac{233}{40})}-\left(-3\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(5.825)}-\log_{5.825}\left(\frac{1600}{37948011}\right)$$

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