How do I solve this?

Question

A condo was purchased for $88,500 in 2000. Its value appreciates 6.5% annually. What will it be

worth in 2015?

Is the equation 88500(6.5)^x

How do I solve it without using (log)??

Answer 1

you are very close!

When dealing with appreciating value, you have to first convert the percentage to a decimal (divide by 100), then add 1. Without adding 1, you'd simply be calculating what 6.5% of $88,500 is. Adding the 1 is basically "6.5% of $85000 plus $85,000", which gives you a bigger number instead of a smaller one.

So our formula is:

y=88500(1.065)^x

You do not have to use the log function here. We are given that we are looking for the value of the condo after 15 years, so x=15

y = 88500(1.065)¹⁵

1.065¹⁵ = 2.572

y = 88500*2.572 = $227,607.93

If you'd like to go over how all of this works in more detail, I'd be happy to help out, just let me know!

Edit:

log is the opposite of an exponential. For example, let's say instead of "what is the value in 2015?", the question instead asked "when will the value of the condo reach $227,607.93?"

We start out with the same formula as before, but instead we plug in $227,607.93 for y.

y=88500(1.065)^x

227607.93 = 88500(1.065)^x

2.572 = 1.065^x This basically says: 1.065 to the power of what equals 2.572

We would solve for x by writing it like this: Log_1.065(2.572) = x

saying this out loud sounds like "log base 1.065 of 2.572 equals x."

in my calculator (ti-84+), there is a "logBASE" function under "MATH". You then type in the numbers just like the formula, and it gives us x = 15.

ln (called natural log) is the same thing, but the base is always e. e is kind of like pi, it's special number that comes up a lot in math to describe exponential growth. e ≈ 2.71828. If you have a graphing calculator, you should have an e^x button and an LN button.

Answer 2

You are almost there with the equation. Since the value appreciates by 6.5%, we know that the condo is increasing in value by 6.5% each year. This means that we need to add 6.5% to the value each year. Since the original value is 100%, if we add 6.5% we get 106.5%. If we convert this to a decimal, we get 1.065; the value of the condo is multiplied by 1.065 each year. Thus, the value (let's call it V) of the condo x years after 2000 is represented by the equation

V = 88,500(1.065)^x

2015 is 15 years after 2000, so we input 15 as x and solve:

V = 88,500(1.065)¹⁵ = $227,607.93

Answer 3

Let V(t) = value of the condo t years after 2000.

The, V(t) = 88,500(1 + 0.065)^t

Value in the year 2015 is V(15) = 88,500(1.065)¹⁵