Solve 5^5x=3

$${ 5 }^{ 5x } =3$$

Solve for x

$$x=\frac{\log_{5}\left(3\right)}{5}\approx 0.136521239$$

Steps Using Definition of Logarithm

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

$$5^{5x}=3$$

Take the logarithm of both sides of the equation.

$$\log(5^{5x})=\log(3)$$

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

$$5x\log(5)=\log(3)$$

Divide both sides by \log(5).

$$5x=\frac{\log(3)}{\log(5)}$$

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

$$5x=\log_{5}\left(3\right)$$

Divide both sides by 5.

$$x=\frac{\log_{5}\left(3\right)}{5}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{5\ln(5)}+\frac{\log_{5}\left(3\right)}{5}$$

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