Solve 511=2^n-1

$$511 = 2 ^ { n - 1 }$$

Solve for n

$$n=\log_{2}\left(511\right)+1\approx 9.997179481$$

Steps Using Definition of Logarithm

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

$$2^{n-1}=511$$

Take the logarithm of both sides of the equation.

$$\log(2^{n-1})=\log(511)$$

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

$$\left(n-1\right)\log(2)=\log(511)$$

Divide both sides by \log(2).

$$n-1=\frac{\log(511)}{\log(2)}$$

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

$$n-1=\log_{2}\left(511\right)$$

Add 1 to both sides of the equation.

$$n=\log_{2}\left(511\right)-\left(-1\right)$$