MHB Solve Logarithms Problems: Find Profit & Interest Rate

[Yankel]

Hello all,

I have a few small questions regarding logarithms, which I would like to ask your help with.

1) A car loses 50% of it's value every 4 years. How many years does it take for the car to lose 1/3 of it's value ?

(I think you need log with basis of 2 here, but not sure)

2) A company sell an app for phones. In the first 6 months of the year the price goes up by 10% a month. During July, August and September the price went down by 5% each month. During October, November and December it went down by 8% each month. At the end of the year, did the company made profit or lost ? Given an answer in percentages. What is the average monthly interest rate ? Find a function and only then a numerical approximation.

Thank you !

In (2) can I say that the yearly price is:

\[f(x)=1.1^{6}0.95^{3}0.92^{3}x\]

where x is the price of the app ?

 

[Greg]

1) An expression for the value of the car after $t$ years is $\left(\frac12\right)^{t/4}\cdot V$, where $V$ is the initial value of the car.Solve $\left(\frac12\right)^{t/4}=\frac13$ for $t$. Can you continue?

 

[Yankel]

I am not sure I am with you on this one. The question is about a third of the initial value, why is it just 1/3 and not 1/3 out of V? The number according to your formula is 6.34, but I am not sure where the formula came from.

 

[HOI]

"1) A car loses 50% of it's value every 4 years. How many years does it take for the car to lose 1/3 of it's value ?"
In t years, there are t/4 4 year periods. So in t years, the cars value will have been multiplied by 1/2 t/4 times. Multiplying by 1/2 t/4 times gives [math]\left(\frac{1}{2}\right)^{\frac{t}{4}}[/math]. If the cars initial value was V, after t years, its value will be [math]\left(\frac{1}{2}\right)^{\frac{t}{4}}V[/math]. We asked to find t when that will be 1/3 the cars value, [math]\frac{1}{3}V[/math]. So we want to find t that satisfies [math]\left(\frac{1}{2}\right)^{\frac{t}{4}}V= \frac{1}{3}V[/math]

Now, DIVIDE both sides of $\left(\frac{1}{2}\right)^{\frac{t}{4}}V= \frac{1}{3}V$ by V!

Your second question makes no sense at all! If all you are given is what percentage the price went up or down, with no information about the actual price or what it cost to create the app to begin with, there is no possible way to calculate the companies profit or loss! The problem then asks for the monthly interest rate! What interest rate? Nothing is said about interest in the problem.