{"id":275,"date":"2023-06-21T13:22:51","date_gmt":"2023-06-21T13:22:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/putting-it-together-exponential-and-logarithmic-functions\/"},"modified":"2023-07-09T04:01:57","modified_gmt":"2023-07-09T04:01:57","slug":"putting-it-together-exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/putting-it-together-exponential-and-logarithmic-functions\/","title":{"raw":"Putting It Together: Exponential and Logarithmic Functions","rendered":"Putting It Together: Exponential and Logarithmic Functions"},"content":{"raw":"At the start of this module, you were considering investing your inheritance possibly to save for retirement. Now you can use what you\u2019ve learned to figure it out. The final value of your investment can be represented by the equation\r\n

[latex]f(t)=Pe^{\\large{rt}}[\/latex]<\/p>\r\nwhere\r\n

[latex]P[\/latex] = the initial investment<\/p>\r\n

[latex]t[\/latex] = number of years invested<\/p>\r\n

[latex]r[\/latex] = interest rate, expressed as a decimal<\/p>\r\nNow remember that you had $10,000 to invest, so [latex]P=10,000[\/latex]. Also recall that the interest rate was 3%, so [latex]r=0.03[\/latex].\r\n\r\nLet\u2019s start with 5 years, so [latex]t=5[\/latex].\r\n

\r\n\r\n\r\n\r\n\r\n\r\n
Start with the function:<\/td>\r\n[latex]f(t)=Pe^{\\large{rt}}[\/latex]<\/td>\r\n<\/tr>\r\n
Substitute P, r, and t:<\/td>\r\n[latex]f(5)=10,000e^{\\large{0.03}{(5)}}[\/latex]<\/td>\r\n<\/tr>\r\n
Evaluate:<\/td>\r\n[latex]f(5)=11,618.34[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n

Now let\u2019s look at 10 years, so [latex]t= 10[\/latex].<\/p>\r\n\r\n

\r\n\r\n\r\n\r\n\r\n\r\n
Start with the function:<\/td>\r\n[latex]f(t)=Pe^{\\large{rt}}[\/latex]<\/td>\r\n<\/tr>\r\n
Substitute P, r, and t:<\/td>\r\n[latex]f(10)=10,000e^{\\large{0.03}{(10)}}[\/latex]<\/td>\r\n<\/tr>\r\n
Evaluate:<\/td>\r\n[latex]f(10)=13,498.59[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow let\u2019s look at 50 years, so [latex]t=50[\/latex].\r\n
\r\n\r\n\r\n\r\n\r\n\r\n
Start with the function:<\/td>\r\n[latex]f(t)=Pe^{\\large{tr}}[\/latex]<\/td>\r\n<\/tr>\r\n
Substitute P, r, and t:<\/td>\r\n[latex]f(50)=10,000e^{\\large{0.03}{(50)}}[\/latex]<\/td>\r\n<\/tr>\r\n
Evaluate:<\/td>\r\n[latex]f(10)=44,816.89[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nUsing the function for continuously compounded interest, you can see how your initial investment will grow over time.\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex]<\/td>\r\nInterest rate<\/td>\r\n[latex]f(t)[\/latex]<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n0.03<\/td>\r\n$11,618.34<\/td>\r\n<\/tr>\r\n
10<\/td>\r\n0.03<\/td>\r\n$13.498.59<\/td>\r\n<\/tr>\r\n
50<\/td>\r\n0.03<\/td>\r\n$44,816.89<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNow you know that your $10,000 can grow to over $44,000 in 50 years! With that knowledge under your belt, you can decide if you want to add to your investment or find an account with a greater interest rate. Either way, thanks to your knowledge of exponential functions, you can make sound financial decisions.","rendered":"

At the start of this module, you were considering investing your inheritance possibly to save for retirement. Now you can use what you\u2019ve learned to figure it out. The final value of your investment can be represented by the equation<\/p>\n

[latex]f(t)=Pe^{\\large{rt}}[\/latex]<\/p>\n

where<\/p>\n

[latex]P[\/latex] = the initial investment<\/p>\n

[latex]t[\/latex] = number of years invested<\/p>\n

[latex]r[\/latex] = interest rate, expressed as a decimal<\/p>\n

Now remember that you had $10,000 to invest, so [latex]P=10,000[\/latex]. Also recall that the interest rate was 3%, so [latex]r=0.03[\/latex].<\/p>\n

Let\u2019s start with 5 years, so [latex]t=5[\/latex].<\/p>\n

\n\n\n\n\n\n
Start with the function:<\/td>\n[latex]f(t)=Pe^{\\large{rt}}[\/latex]<\/td>\n<\/tr>\n
Substitute P, r, and t:<\/td>\n[latex]f(5)=10,000e^{\\large{0.03}{(5)}}[\/latex]<\/td>\n<\/tr>\n
Evaluate:<\/td>\n[latex]f(5)=11,618.34[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Now let\u2019s look at 10 years, so [latex]t= 10[\/latex].<\/p>\n

\n\n\n\n\n\n
Start with the function:<\/td>\n[latex]f(t)=Pe^{\\large{rt}}[\/latex]<\/td>\n<\/tr>\n
Substitute P, r, and t:<\/td>\n[latex]f(10)=10,000e^{\\large{0.03}{(10)}}[\/latex]<\/td>\n<\/tr>\n
Evaluate:<\/td>\n[latex]f(10)=13,498.59[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Now let\u2019s look at 50 years, so [latex]t=50[\/latex].<\/p>\n

\n\n\n\n\n\n
Start with the function:<\/td>\n[latex]f(t)=Pe^{\\large{tr}}[\/latex]<\/td>\n<\/tr>\n
Substitute P, r, and t:<\/td>\n[latex]f(50)=10,000e^{\\large{0.03}{(50)}}[\/latex]<\/td>\n<\/tr>\n
Evaluate:<\/td>\n[latex]f(10)=44,816.89[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Using the function for continuously compounded interest, you can see how your initial investment will grow over time.<\/p>\n

\n\n\n\n\n\n\n
[latex]t[\/latex]<\/td>\nInterest rate<\/td>\n[latex]f(t)[\/latex]<\/td>\n<\/tr>\n
5<\/td>\n0.03<\/td>\n$11,618.34<\/td>\n<\/tr>\n
10<\/td>\n0.03<\/td>\n$13.498.59<\/td>\n<\/tr>\n
50<\/td>\n0.03<\/td>\n$44,816.89<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

Now you know that your $10,000 can grow to over $44,000 in 50 years! With that knowledge under your belt, you can decide if you want to add to your investment or find an account with a greater interest rate. Either way, thanks to your knowledge of exponential functions, you can make sound financial decisions.<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Original<\/div>