Solve 0.5^frac{x{5760}}=0.153

$$0.5 ^ { \frac { x } { 5760 } } = 0.153$$

Solve for x

$$x=\log_{2}\left(\frac{5^{17280}}{153^{5760}}\right)+17280$$

Steps Using Definition of Logarithm

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

$$0.5^{\frac{1}{5760}x}=0.153$$

Take the logarithm of both sides of the equation.

$$\log(0.5^{\frac{1}{5760}x})=\log(0.153)$$

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

$$\frac{1}{5760}x\log(0.5)=\log(0.153)$$

Divide both sides by \log(0.5).

$$\frac{1}{5760}x=\frac{\log(0.153)}{\log(0.5)}$$

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

$$\frac{1}{5760}x=\log_{0.5}\left(0.153\right)$$

Multiply both sides by 5760.

$$x=-\frac{\frac{\ln(\frac{153}{1000})}{\ln(2)}}{\frac{1}{5760}}$$