Solve 1=512(1/2)^frac{m{45}}

$$1 = 512 ( \frac { 1 } { 2 } ) ^ { \frac { m } { 45 } }$$

Solve for m

$$m=405$$

Steps Using Definition of Logarithm

Divide both sides by 512.

$$\frac{1}{512}=\left(\frac{1}{2}\right)^{\frac{m}{45}}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{1}{2}\right)^{\frac{m}{45}}=\frac{1}{512}$$

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

$$\left(\frac{1}{2}\right)^{\frac{1}{45}m}=\frac{1}{512}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{\frac{1}{45}m})=\log(\frac{1}{512})$$

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

$$\frac{1}{45}m\log(\frac{1}{2})=\log(\frac{1}{512})$$

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

$$\frac{1}{45}m=\frac{\log(\frac{1}{512})}{\log(\frac{1}{2})}$$

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

$$\frac{1}{45}m=\log_{\frac{1}{2}}\left(\frac{1}{512}\right)$$

Multiply both sides by 45.

$$m=\frac{9}{\frac{1}{45}}$$