Solve 178000/81000={left(1+.064/365right)}^365t

$$\frac{ 178000 }{ 81000 } = { \left(1+ \frac{ .064 }{ 365 } \right) }^{ 365t }$$

Solve for t

$$t = \frac{\log_{\frac{45633}{45625}} {(\frac{178}{81})}}{365} \approx 12.30317844$$

Steps Using Definition of Logarithm

Reduce the fraction \frac{178000}{81000} to lowest terms by extracting and canceling out 1000.

$$\frac{178}{81}=\left(1+\frac{0.064}{365}\right)^{365t}$$

Expand \frac{0.064}{365} by multiplying both numerator and the denominator by 1000.

$$\frac{178}{81}=\left(1+\frac{64}{365000}\right)^{365t}$$

Reduce the fraction \frac{64}{365000} to lowest terms by extracting and canceling out 8.

$$\frac{178}{81}=\left(1+\frac{8}{45625}\right)^{365t}$$

Add 1 and \frac{8}{45625} to get \frac{45633}{45625}.

$$\frac{178}{81}=\left(\frac{45633}{45625}\right)^{365t}$$

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

$$\left(\frac{45633}{45625}\right)^{365t}=\frac{178}{81}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{45633}{45625}\right)^{365t})=\log(\frac{178}{81})$$

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

$$365t\log(\frac{45633}{45625})=\log(\frac{178}{81})$$

Divide both sides by \log(\frac{45633}{45625}).

$$365t=\frac{\log(\frac{178}{81})}{\log(\frac{45633}{45625})}$$

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

$$365t=\log_{\frac{45633}{45625}}\left(\frac{178}{81}\right)$$

Divide both sides by 365.

$$t=\frac{\ln(\frac{178}{81})}{365\ln(\frac{45633}{45625})}$$

Solve for t (complex solution)

$$t=\frac{i\times 2\pi n_{1}}{365\ln(\frac{45633}{45625})}-\frac{\log_{\frac{45633}{45625}}\left(\frac{81}{178}\right)}{365}$$

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