Solve 5=(1+9.6%)^n

$$5 = ( 1 + 9.6 \% ) ^ { n }$$

Solve for n

$$n=\log_{1.096}\left(5\right)\approx 17.557404545$$

Steps Using Definition of Logarithm

Expand \frac{9.6}{100} by multiplying both numerator and the denominator by 10.

$$5=\left(1+\frac{96}{1000}\right)^{n}$$

Reduce the fraction \frac{96}{1000} to lowest terms by extracting and canceling out 8.

$$5=\left(1+\frac{12}{125}\right)^{n}$$

Add 1 and \frac{12}{125} to get \frac{137}{125}.

$$5=\left(\frac{137}{125}\right)^{n}$$

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

$$\left(\frac{137}{125}\right)^{n}=5$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{137}{125}\right)^{n})=\log(5)$$

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

$$n\log(\frac{137}{125})=\log(5)$$

Divide both sides by \log(\frac{137}{125}).

$$n=\frac{\log(5)}{\log(\frac{137}{125})}$$

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

$$n=\log_{\frac{137}{125}}\left(5\right)$$