#### Answer

After $15$ years, the investment will double in value.

#### Work Step by Step

Interest is earned at the rate of $4.75\%$ compounded annually means that every year, the investment comes out as $104.75\%$ of the amount of the previous year.
For example, here we have $\$500$ initially. After the first year, the investment values $104.75\%$ of $\$500$; in other words, its value now is $\$500\times1.0475$.
After the second year, it values $104.75\%$ the investment of the previous year, which means its value now is $1.0475\times\$500\times1.0475=\$500\times(1.0475)^2$
Therefore, if we continue like that and call the value of the investment after $t$ years $y$, we can come up with the model to calculate that amount:
$$y=500\times1.0475^t$$
For an investment of $\$500$ to double in value, meaning to find $t$ so that $y=1000$:
$$500\times1.0475^t=1000$$ $$1.0475^t=2$$
- Take the $\log_{1.0475}$ of both sides: $$t=\log_{1.0475}2\approx14.936\approx15(years)$$