Today, as you were listening to music, did you think about the intensity or loudness measured in decibels (db) of the sound? Decibel intensity can be modeled using a logarithmic equation. One such equation that can model this is d = 10 log101 over 2 l l0 , which is a logarithmic relationship.
The intensity or decibel level that you find reasonable would probably not be reasonable to a physician. The reasonableness of the situation depends on who is making that decision.
Set music to the side for a moment, and suppose that you are thinking about buying a car. Right now, you may not have enough income or savings to purchase the car of your dreams, which may cost you at least $15,000. However, there are many ways to save to purchase the car.
One way that you can save money is to invest it in an account where the interest is compounded annually. The function below will calculate the amount saved, S(t), when $5000 is compounded annually for t years at an interest rate of r.
How long will it take to reach your goal of $15,000 if the annual interest rate is 5%? Before solving the problem, think about a reasonable amount of time to wait to purchase your car.
First, set up the problem as follows:
S(t) = 5000(1 + 0.05)t
S(t) = 5000(1.05)t
You are probably thinking, where is the logarithm? In order to solve for t, you need to find the inverse of the function or the logarithmic function. Below is the actual calculation for this problem.
Click through the animation below to view one student’s solution to the problem.
Reflecting on the solution, is this a reasonable amount of time for you to wait? Use the interactive below to look at several different options that use similar equations to see if this would be a reasonable way to save for a car.
Using the equation above, match the equation with a reasonable time associated with the equation.
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The two values that would allow you to save the most money in the shortest time are the interest rate and the principle. Starting with a greater amount of money and investing it at a higher interest rate would produce a larger amount of savings more quickly.Interactive popup. Assistance may be required.
One reasonable answer is having the $15,000 returned as quickly as possible. In that case, you would probably want them to give you $10,000 with a 7.25% interest rate because then you would be reimbursed within 6 years.So far, you have evaluated compound interest over a reasonable time to obtain a desired income. Compound interest will accumulate a larger income than simple interest, but it isn’t always a reasonable choice.
You would like to take your family on a luxury vacation. If you invested $500 at 6.5% when you are 18, when would you be able to take your family on the luxury vacation if it cost $25,000? What would be a reasonable amount of time to save for the vacation and why?
S(t) = P(1 + r)t
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Comparing this situation with the one above, it might take too long to earn the desired amount.In the problem where you saved for a car, it took 18 years for $5,000 to grow to $15,000 at a slightly lower rate. If you started with $500, it would take a lot longer to save for the vacation this way. Below is the actual solution:
25,000 = 500(1.065)t
25,000 over 500
25,000
500
=
500 over 500
500
500
(1.065)t
50 = (1.065)t
log50 = log(1.065)t
log50 = log(1.065)
log50 over log(1.065)
log50
log(1.065)
=
tlog(1.065) over log(1.065)
log(1.065)
log(1.065)
1.70 over 0.027
1.70
0.027
= t ≈ 63 years
Sometimes the solution to a logarithmic equation is not algebraically reasonable. Look at the example below, click on the equation to see the next step.
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Follow the steps above to solve the problem.Interactive popup. Assistance may be required.
One solution, x = -1 is not a reasonable solution, it is not possible to take the logarithm of a negative number.Interactive popup. Assistance may be required.
Follow the steps above to solve the problem but change the base of the logarithm to 10.Interactive popup. Assistance may be required.
One solution, x = -10, is not a reasonable solution because when the value of -10 is substituted into the equation, it is not possible to take the logarithm of a negative number.Interactive popup. Assistance may be required.
Follow the steps above to solve the problem. However, this time, it is a natural log (the inverse function of e, e = 2.71828. . .) or the base is e.Interactive popup. Assistance may be required.
In(x − 2) + In(x − 2) = 0Interactive popup. Assistance may be required.
Set up the equation with the given values, and review the compound interest problems noting the interest rate.Interactive popup. Assistance may be required.
It is not reasonable to expect to double the principal in 5 years at a rate of 3.25%.Interactive popup. Assistance may be required.
Review both methods presented in the lesson.Interactive popup. Assistance may be required.
You would earn income more quickly compounding interest continually, since interest is compounded on the given amount.