Solve 2500=1000{left(1+12/2.8right)}^12x

$$2500 = 1000 { \left(1+ \frac{ 12 }{ 2.8 } \right) }^{ 12x }$$

Solve for x

$$x=\frac{\log_{\frac{37}{7}}\left(2.5\right)}{12}\approx 0.045860183$$

Steps Using Definition of Logarithm

Divide both sides by 1000.

$$\frac{2500}{1000}=\left(1+\frac{12}{2.8}\right)^{12x}$$

Reduce the fraction \frac{2500}{1000} to lowest terms by extracting and canceling out 500.

$$\frac{5}{2}=\left(1+\frac{12}{2.8}\right)^{12x}$$

Expand \frac{12}{2.8} by multiplying both numerator and the denominator by 10.

$$\frac{5}{2}=\left(1+\frac{120}{28}\right)^{12x}$$

Reduce the fraction \frac{120}{28} to lowest terms by extracting and canceling out 4.

$$\frac{5}{2}=\left(1+\frac{30}{7}\right)^{12x}$$

Add 1 and \frac{30}{7} to get \frac{37}{7}.

$$\frac{5}{2}=\left(\frac{37}{7}\right)^{12x}$$

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

$$\left(\frac{37}{7}\right)^{12x}=\frac{5}{2}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{37}{7}\right)^{12x})=\log(\frac{5}{2})$$

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

$$12x\log(\frac{37}{7})=\log(\frac{5}{2})$$

Divide both sides by \log(\frac{37}{7}).

$$12x=\frac{\log(\frac{5}{2})}{\log(\frac{37}{7})}$$

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

$$12x=\log_{\frac{37}{7}}\left(\frac{5}{2}\right)$$

Divide both sides by 12.

$$x=\frac{\ln(\frac{5}{2})}{12\ln(\frac{37}{7})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{6\ln(\frac{37}{7})}-\frac{\log_{\frac{37}{7}}\left(0.4\right)}{12}$$

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