Solve 995=25{left(1-1+0.0335/12/frac{0.0355{12}}right)}^-n

$$995 = 25 { \left( \frac{ 1-1+ \frac{ 0.0335 }{ 12 } }{ \frac{ 0.0355 }{ 12 } } \right) }^{ -n }$$

Solve for n

$$n=\log_{\frac{71}{67}}\left(39.8\right)\approx 63.52890379$$

Steps Using Definition of Logarithm

Divide both sides by 25.

$$\frac{995}{25}=\left(\frac{1-1+\frac{0.0335}{12}}{\frac{0.0355}{12}}\right)^{-n}$$

Reduce the fraction \frac{995}{25} to lowest terms by extracting and canceling out 5.

$$\frac{199}{5}=\left(\frac{1-1+\frac{0.0335}{12}}{\frac{0.0355}{12}}\right)^{-n}$$

Subtract 1 from 1 to get 0.

$$\frac{199}{5}=\left(\frac{\frac{0.0335}{12}}{\frac{0.0355}{12}}\right)^{-n}$$

Expand \frac{0.0335}{12} by multiplying both numerator and the denominator by 10000.

$$\frac{199}{5}=\left(\frac{\frac{335}{120000}}{\frac{0.0355}{12}}\right)^{-n}$$

Reduce the fraction \frac{335}{120000} to lowest terms by extracting and canceling out 5.

$$\frac{199}{5}=\left(\frac{\frac{67}{24000}}{\frac{0.0355}{12}}\right)^{-n}$$

Expand \frac{0.0355}{12} by multiplying both numerator and the denominator by 10000.

$$\frac{199}{5}=\left(\frac{\frac{67}{24000}}{\frac{355}{120000}}\right)^{-n}$$

Reduce the fraction \frac{355}{120000} to lowest terms by extracting and canceling out 5.

$$\frac{199}{5}=\left(\frac{\frac{67}{24000}}{\frac{71}{24000}}\right)^{-n}$$

Divide \frac{67}{24000} by \frac{71}{24000} by multiplying \frac{67}{24000} by the reciprocal of \frac{71}{24000}.

$$\frac{199}{5}=\left(\frac{67}{24000}\times \frac{24000}{71}\right)^{-n}$$

Multiply \frac{67}{24000} and \frac{24000}{71} to get \frac{67}{71}.

$$\frac{199}{5}=\left(\frac{67}{71}\right)^{-n}$$

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

$$\left(\frac{67}{71}\right)^{-n}=\frac{199}{5}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{67}{71}\right)^{-n})=\log(\frac{199}{5})$$

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

$$-n\log(\frac{67}{71})=\log(\frac{199}{5})$$

Divide both sides by \log(\frac{67}{71}).

$$-n=\frac{\log(\frac{199}{5})}{\log(\frac{67}{71})}$$

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

$$-n=\log_{\frac{67}{71}}\left(\frac{199}{5}\right)$$

Divide both sides by -1.

$$n=\frac{\ln(\frac{199}{5})}{-\ln(\frac{67}{71})}$$