If I decide to invest $$10,000$ in a mutual fund retirement account with a fixed total annual return of $7\%$ and the broker charges a $2\%$ commission, I can use the compound interest formula to assess the impact of the $2\%$ commission rate over $10$ and $50$ years. To analyze the impact of the $2\%$ commission on investment returns over different periods using the compound interest formula, we need to compare the future values of the investment with $5\%$ and $7\%$ annual returns.

Furthermore, I can compare the growth of the principal over $50$ years at $5\%$ and $7\%.$

By calculating and contrasting the difference over $10$ years, I can examine the values and percentages at the $10$ and $50-$year marks and draw conclusions about their implications.