Exponential Functions Word Problem

[mtiller1]

An unscrupulous title pawn company charges 5% interest per week. Assume that the interest is compounded weekly. Suppose that you borrow $1000 from this title pawn company. How many weeks will it be before you owe them $10,000?

I started it this way:

A(t) = P(1+r/n)^nt

10000 = 1000(1 + 0.05/52)^52t

10000 = 1000.961538^52t

I need to solve for t ... but now I'm stuck.

Please let me know if I have anything else wrong as well.

Thanks in advance!!
Angel

 

[Guest]

I assume you have learnt logs.....

rewrite your equation in log form and then change to a log base 10,

10000 = 1000.961538^52t


52t = log 10000 / log 1000.961538

and solve this

 

[tkhunny]

An unscrupulous title pawn company charges 5% interest per week. Assume that the interest is compounded weekly. Suppose that you borrow $1000 from this title pawn company. How many weeks will it be before you owe them $10,000?

I started it this way:

A(t) = P(1+r/n)^nt

You are substantially confused, here. The problem gives you weekly interest. Why are you dividing it by anything?

10000 = 1000(1 + 0.05/52)^52t

This should be 10000 = 1000*(1.05)^t

10000 = 1000.961538^52t

Yikes! That is VERY unfortunate. Have you encountered the "Order of Operations"? You do not get to do that multiplication, yet. The exponent comes first.

10000 = 1000*(1.05)^t

Logarithms are the way to go.

log(10000) = log(1000*(1.05)^t)
log(10000) = log(1000) + t*log(1.05)
4 = 3 + t*log(1.05)
1 = t*log(1.05)
t = 1/log(1.05)

 

[Denis]

log(10000) = log(1000*(1.05))
log(10000) = log(1000) + t*log(1.05)
4 = 3 + t*log(1.05)
1 = t*log(1.05)
t = 1/log(1.05)

Methinks you erred gravely, TK; t= log(10) / log(1.05), mais oui?

 

[mtiller1]

Thank you all -- tremendously!!! Just a couple follow ups --

You are substantially confused, here. The problem gives you weekly interest. Why are you dividing it by anything?

Yep, that was an easy given -- completely didn't see it!

Quote:
10000 = 1000.961538^52t
Yikes! That is VERY unfortunate. Have you encountered the "Order of Operations"? You do not get to do that multiplication, yet. The exponent comes first.

I thought parenthesis came first -- please excuse my dear aunt sally -- am I completely lost here? (Even though I don't need to divide by 52, I had 10000 = 1000(1 + 0.05/52)^52t before this step. Pretending that was correct, wouldn't I do the parenthesis first?)

Methinks you erred gravely, TK; t= log(10) / log(1.05), mais oui?

Is it "more correct" to use log(10) instead of 1? Doesn't log(10) = 1?

Thanks again! It really helped me out!
Angel

 

[Denis]

log(10) = 2.30258~

That'll give you the number of weeks = 47.19363~

In other words, 1000(1.05)^47.19363~ = 10000

 

[Unco]

log(10000) = log(1000*(1.05))
log(10000) = log(1000) + t*log(1.05)
4 = 3 + t*log(1.05)
1 = t*log(1.05)
t = 1/log(1.05)

Methinks you erred gravely, TK; t= log(10) / log(1.05), mais oui?

"log" without a stated base tends to be assumed to be of base 10.

In your other post you have treated "log" to mean "natural log", which is usually referred to as "ln", but ultimately it doesnt matter because any base could have been chosen.

 

[tkhunny]

I thought it relatively obvious that I was using Base 10 logs.

 

[Denis]

Ya; I shudda noticed; hope I'm forgiven