Solve 14,730=30(7(1+0.335/12)/frac{0.335{12}})^n

$$14,730 = 30 ( \frac { 7 ( 1 + \frac { 0.335 } { 12 } ) } { \frac { 0.335 } { 12 } } ) ^ { n }$$

Solve for n

$$n=\log_{\frac{17269}{67}}\left(491\right)\approx 1.116079126$$

Steps Using Definition of Logarithm

Divide both sides by 30.

$$\frac{14730}{30}=\left(\frac{7\left(1+\frac{0.335}{12}\right)}{\frac{0.335}{12}}\right)^{n}$$

Divide 14730 by 30 to get 491.

$$491=\left(\frac{7\left(1+\frac{0.335}{12}\right)}{\frac{0.335}{12}}\right)^{n}$$

Expand \frac{0.335}{12} by multiplying both numerator and the denominator by 1000.

$$491=\left(\frac{7\left(1+\frac{335}{12000}\right)}{\frac{0.335}{12}}\right)^{n}$$

Reduce the fraction \frac{335}{12000} to lowest terms by extracting and canceling out 5.

$$491=\left(\frac{7\left(1+\frac{67}{2400}\right)}{\frac{0.335}{12}}\right)^{n}$$

Add 1 and \frac{67}{2400} to get \frac{2467}{2400}.

$$491=\left(\frac{7\times \frac{2467}{2400}}{\frac{0.335}{12}}\right)^{n}$$

Multiply 7 and \frac{2467}{2400} to get \frac{17269}{2400}.

$$491=\left(\frac{\frac{17269}{2400}}{\frac{0.335}{12}}\right)^{n}$$

Expand \frac{0.335}{12} by multiplying both numerator and the denominator by 1000.

$$491=\left(\frac{\frac{17269}{2400}}{\frac{335}{12000}}\right)^{n}$$

Reduce the fraction \frac{335}{12000} to lowest terms by extracting and canceling out 5.

$$491=\left(\frac{\frac{17269}{2400}}{\frac{67}{2400}}\right)^{n}$$

Divide \frac{17269}{2400} by \frac{67}{2400} by multiplying \frac{17269}{2400} by the reciprocal of \frac{67}{2400}.

$$491=\left(\frac{17269}{2400}\times \frac{2400}{67}\right)^{n}$$

Multiply \frac{17269}{2400} and \frac{2400}{67} to get \frac{17269}{67}.

$$491=\left(\frac{17269}{67}\right)^{n}$$

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

$$\left(\frac{17269}{67}\right)^{n}=491$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{17269}{67}\right)^{n})=\log(491)$$

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

$$n\log(\frac{17269}{67})=\log(491)$$

Divide both sides by \log(\frac{17269}{67}).

$$n=\frac{\log(491)}{\log(\frac{17269}{67})}$$

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

$$n=\log_{\frac{17269}{67}}\left(491\right)$$