Solve 765=frac{384(frac{1}2^n-1)}{-1/2}

$$765 = \frac { 384 ( \frac { 1 } { 2 } ^ { n } - 1 ) } { - \frac { 1 } { 2 } }$$

Solve for n

$$n=8$$

Steps Using Definition of Logarithm

Multiply both sides by -\frac{1}{2}.

$$765\left(-\frac{1}{2}\right)=384\left(\left(\frac{1}{2}\right)^{n}-1\right)$$

Multiply 765 and -\frac{1}{2} to get -\frac{765}{2}.

$$-\frac{765}{2}=384\left(\left(\frac{1}{2}\right)^{n}-1\right)$$

Use the distributive property to multiply 384 by \left(\frac{1}{2}\right)^{n}-1.

$$-\frac{765}{2}=384\times \left(\frac{1}{2}\right)^{n}-384$$

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

$$384\times \left(\frac{1}{2}\right)^{n}-384=-\frac{765}{2}$$

Add 384 to both sides of the equation.

$$384\times \left(\frac{1}{2}\right)^{n}=\frac{3}{2}$$

Divide both sides by 384.

$$\left(\frac{1}{2}\right)^{n}=\frac{1}{256}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{2}\right)^{n})=\log(\frac{1}{256})$$

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

$$n\log(\frac{1}{2})=\log(\frac{1}{256})$$

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

$$n=\frac{\log(\frac{1}{256})}{\log(\frac{1}{2})}$$

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

$$n=\log_{\frac{1}{2}}\left(\frac{1}{256}\right)$$

Solve for n (complex solution)

$$n=-\frac{2\pi n_{1}i}{\ln(2)}+8$$

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