1. Exponential Modeling (Compound Interest) Logs and Exponents Question 4 Solution

2. Solution To approach this kind of question you must bring in the compound interest formula, which is: Formula: A = P (1 + r ) t n n A = the final amount. P = Principle or present value of the final amount A . r = the rate, which should always be expressed as a decimal. (ex: 4% would be 0.04.) This formula is used when you know 3 of the variables. When three are known, you can find out what the fourth variable is. n = number of compounding periods per year. t = period or number of years

3. Plug in all the information you know given from the question. A = 1 000 000 P = 10 000 r = 10% ∴ divide 10 by 100 to get it in decimal form. which is 0.1 n = 1 since the interest is being compounded annually. In other words it's being compounded once yearly. t = unknown. This is what we're trying to find. It's all in the basics of isolating the t . Wait! we're not done yet! go onto the next page to continue... Formula: A = P (1 + r ) t n n 1 000 000 10 000 = 10 000 (1.01) t 10 000 100 = (1.01) t 1 000 000 = 10 000 (1 + 0.1 ) t ( 1 ) 1

4. Since t is an exponent, we can't just divide both sides by 1.01. We must first make the bases the same by taking the log of both sides. When no base is indicated, it is base 10, the "Common Log". When the bases are the same we can then use the "power law" to bring down the exponent. (Taking the base 10 of both sides is much easier than making the bases a 2 or a 3 since you can put it in your calculator.) Now we can properly isolate the t. 100 (1.01) t = log log 100 = log (1.01) log 100 = log (1.01) log log log (1.01) (1.01) t t

5. FINAL ANSWER It would take approximately 48 years for Shinobi to invest up to $1 000 000. 100 (1.01) = log log t t ≈ 48.3177 t ≈ 48 years