Solve 1022=2(2^n-1)/2-1

$$1022 = \frac { 2 ( 2 ^ { n } - 1 ) } { 2 - 1 }$$

Solve for n

$$n=9$$

Steps Using Definition of Logarithm

Subtract 1 from 2 to get 1.

$$1022=\frac{2\left(2^{n}-1\right)}{1}$$

Anything divided by one gives itself.

$$1022=2\left(2^{n}-1\right)$$

Use the distributive property to multiply 2 by 2^{n}-1.

$$1022=2\times 2^{n}-2$$

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

$$2\times 2^{n}-2=1022$$

Add 2 to both sides of the equation.

$$2\times 2^{n}=1024$$

Divide both sides by 2.

$$2^{n}=512$$

Take the logarithm of both sides of the equation.

$$\log(2^{n})=\log(512)$$

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

$$n\log(2)=\log(512)$$

Divide both sides by \log(2).

$$n=\frac{\log(512)}{\log(2)}$$

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

$$n=\log_{2}\left(512\right)$$