Solve {left(1/16right)}^x=2^5

$${ \left( \frac{ 1 }{ 16 } \right) }^{ x } = { 2 }^{ 5 }$$

Solve for x

$$x = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

Steps Using Definition of Logarithm

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

$$\left(\frac{1}{16}\right)^{x}=32$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{16}\right)^{x})=\log(32)$$

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

$$x\log(\frac{1}{16})=\log(32)$$

Divide both sides by \log(\frac{1}{16}).

$$x=\frac{\log(32)}{\log(\frac{1}{16})}$$

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

$$x=\log_{\frac{1}{16}}\left(32\right)$$

Solve for x (complex solution)

$$x=-\frac{\pi n_{1}i}{2\ln(2)}-\frac{5}{4}$$

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