Solve 3^frac{x{5}}=37

$$3 ^ { \frac { x } { 5 } } = 37$$

Solve for x

$$x=5\log_{3}\left(37\right)\approx 16.433995641$$

Steps Using Definition of Logarithm

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

$$3^{\frac{1}{5}x}=37$$

Take the logarithm of both sides of the equation.

$$\log(3^{\frac{1}{5}x})=\log(37)$$

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

$$\frac{1}{5}x\log(3)=\log(37)$$

Divide both sides by \log(3).

$$\frac{1}{5}x=\frac{\log(37)}{\log(3)}$$

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

$$\frac{1}{5}x=\log_{3}\left(37\right)$$

Multiply both sides by 5.

$$x=\frac{\log_{3}\left(37\right)}{\frac{1}{5}}$$

Solve for x (complex solution)

$$x=\frac{10\pi n_{1}i}{\ln(3)}+5\log_{3}\left(37\right)$$

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