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.

Conclusion Questions

• Review the matches above. If you wanted to earn the most income possible in the shortest amount of time, which part or parts of the equation would help you select the best option to select?

  Interactive popup. Assistance may be required.

  Check Your Answer

  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.

• Imagine you are loaning someone $15,000. Which of the above equations would you select to determine how you will be repaid and why?

  Interactive popup. Assistance may be required.

  Check Your Answer

  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.

Pause and Reflect

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

Interactive popup. Assistance may be required.

Check Your Answer

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.

Practice

1. Decide whether the solutions for logx + log(x – 3) = 0 are both reasonable solutions. Explain why or why not.

   Interactive popup. Assistance may be required.

   Need a hint?

   Follow the steps above to solve the problem.

   Interactive popup. Assistance may be required.

   Check Your Answer

   One solution, x = -1 is not a reasonable solution, it is not possible to take the logarithm of a negative number.
   
   log₂x + log₂(x – 3) = 2
   log₂[x(x – 3)] = 2
   2^{log₂[x(x – 3)]} = 2²
   x(x – 3) = 4
   x² – 3x – 4 = 0
   (x + 1)(x – 4) = 0
   x = -1 or x = 4

2. Decide whether the solutions of log₁₀(x) + log₁₀(x + 9) = 1 are both reasonable solutions. Explain why or why not.

   Interactive popup. Assistance may be required.

   Need a hint?

   Follow the steps above to solve the problem but change the base of the logarithm to 10.

   Interactive popup. Assistance may be required.

   Check Your Answer

   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.
   
   log₁₀x + log₁₀(x + 9) = 1
   log₁₀[x(x + 9)] = 1
   10^{log₁₀[x(x + 9)]} = 10¹
   x(x + 9) = 10
   x² + 9x – 10 = 0
   (x + 10)(x – 1) = 0
   x = -10 or x = 1

3. Decide whether the solutions of log₂x + log₂(x − 3) = 2 are reasonable solutions. Explain why or why not.

   Interactive popup. Assistance may be required.

   Need a hint?

   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.

   Check Your Answer

   In(x − 2) + In(x − 2) = 0
   In[(x − 2)(x − 2)] = 0
   e^{In[(x − 2)(x − 2)]} = e⁰
   (x − 2)(x − 2) = 1
   x² − 4x + 4 = 1
   x² − 4x + 3 = 0
   (x − 3)(x − 1) = 0
   x = 3 or 1
   In(3 − 2) = In exists so x = 3 is reasonable.
   In(1 − 2) = In(-2) does not exist so it is unreasonable.

4. Instead of using the equation for compound interest, institutions often use the formula to compound interest continually, which is of A = Pe^rt where A equals the amount after a given time, P is the initial amount deposited, r is the annual interest rate, and t is the number of years. If $5,250 is deposited in an institution paying an annual interest rate of 3.25%, is it reasonable to expect to double the principal in 5 years?

   Interactive popup. Assistance may be required.

   Need a hint?

   Set up the equation with the given values, and review the compound interest problems noting the interest rate.

   Interactive popup. Assistance may be required.

   Check Your Answer

   It is not reasonable to expect to double the principal in 5 years at a rate of 3.25%.
   
   10,000 = 5000e^0.0325t
   2 = e^0.0325t
   In2 = 0.0325t
   0.6931 = 030625t
   t ≈ 21
   
   OR
   
   A = 5000(e^0.0325(5))
   A = 5000(e^0.1625)
   A = 5000(1.17644)
   A = $5882

5. Would you earn more income quicker using the compound interest formula or the continuous compound interest formula? Why?

   Interactive popup. Assistance may be required.

   Need a hint?

   Review both methods presented in the lesson.

   Interactive popup. Assistance may be required.

   Check Your Answer

   You would earn income more quickly compounding interest continually, since interest is compounded on the given amount.