Solve 2000=1000(1+0.08/12)^12x

$$2000 = 1000 ( 1 + \frac { 0.08 } { 12 } ) ^ { 12 x }$$

Solve for x

$$x = \frac{\log_{\frac{151}{150}} {(2)}}{12} \approx 8.693188906$$

Steps Using Definition of Logarithm

Divide both sides by 1000.

$$\frac{2000}{1000}=\left(1+\frac{0.08}{12}\right)^{12x}$$

Divide 2000 by 1000 to get 2.

$$2=\left(1+\frac{0.08}{12}\right)^{12x}$$

Expand \frac{0.08}{12} by multiplying both numerator and the denominator by 100.

$$2=\left(1+\frac{8}{1200}\right)^{12x}$$

Reduce the fraction \frac{8}{1200} to lowest terms by extracting and canceling out 8.

$$2=\left(1+\frac{1}{150}\right)^{12x}$$

Add 1 and \frac{1}{150} to get \frac{151}{150}.

$$2=\left(\frac{151}{150}\right)^{12x}$$

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

$$\left(\frac{151}{150}\right)^{12x}=2$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{151}{150}\right)^{12x})=\log(2)$$

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

$$12x\log(\frac{151}{150})=\log(2)$$

Divide both sides by \log(\frac{151}{150}).

$$12x=\frac{\log(2)}{\log(\frac{151}{150})}$$

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

$$12x=\log_{\frac{151}{150}}\left(2\right)$$

Divide both sides by 12.

$$x=\frac{\ln(2)}{12\ln(\frac{151}{150})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{6\ln(\frac{151}{150})}+\frac{\log_{\frac{151}{150}}\left(2\right)}{12}$$

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