Solve 36(1/2)^x/5=4

$$36 ( \frac { 1 } { 2 } ) ^ { x / 5 } = 4$$

Solve for x

$$x=10\log_{2}\left(3\right)\approx 15.849625007$$

Steps Using Definition of Logarithm

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

$$36\times \left(\frac{1}{2}\right)^{\frac{1}{5}x}=4$$

Divide both sides by 36.

$$\left(\frac{1}{2}\right)^{\frac{1}{5}x}=\frac{1}{9}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{\frac{1}{5}x})=\log(\frac{1}{9})$$

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

$$\frac{1}{5}x\log(\frac{1}{2})=\log(\frac{1}{9})$$

Divide both sides by \log(\frac{1}{2}).

$$\frac{1}{5}x=\frac{\log(\frac{1}{9})}{\log(\frac{1}{2})}$$

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

$$\frac{1}{5}x=\log_{\frac{1}{2}}\left(\frac{1}{9}\right)$$

Multiply both sides by 5.

$$x=\frac{2\log_{2}\left(3\right)}{\frac{1}{5}}$$