Solve 2^frac{x{3}}=52

$$2 ^ { \frac { x } { 3 } } = 52$$

Solve for x

$$x=3\left(\log_{2}\left(13\right)+2\right)\approx 17.101319154$$

Steps Using Definition of Logarithm

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

$$2^{\frac{1}{3}x}=52$$

Take the logarithm of both sides of the equation.

$$\log(2^{\frac{1}{3}x})=\log(52)$$

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

$$\frac{1}{3}x\log(2)=\log(52)$$

Divide both sides by \log(2).

$$\frac{1}{3}x=\frac{\log(52)}{\log(2)}$$

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

$$\frac{1}{3}x=\log_{2}\left(52\right)$$

Multiply both sides by 3.

$$x=\frac{\log_{2}\left(52\right)}{\frac{1}{3}}$$