Question

I need help. Basically, I need to figure out the dependent and independent value of a function.

Here is the situation thing:
The chess club started with a membership of only 4 students, but the membership grew at a rate of 2.56% each week. The number of students is modeled by f(x) = 4(1.0256)x.
I had to make like a question about it. I chose "In how many weeks would there be 12 students?".
So that made the equation 12 = 4(1.0256)x.
Then I had to use logarithms to get to the answer (I got 43.46, if you could fact check this too, that would be lovely.)

So the question is:
Let your dependent variable in the function be y. Write the function that models the independent variable in terms of y, using logarithms.

I really don't know what it wants me to do, please help

Answer 1

Answer:

• x = log(y/4)/log(1.0256)
• your answer for y=12 is correct

Step-by-step explanation:

The question is asking you to solve ...

   y = f(x)

for x. (In other words, find the inverse function.)

You already did this using a constant for y. Do the same thing with y instead of the constant.

  y = 4(1.0256^x)

  y/4 = 1.0256^x . . . . . . . divide by 4

  log(y/4) = x·log(1.0256) . . . . . take logs

  log(y/4)/log(1.0256) = x . . . . . divide by the coefficient of x

Now, you have a model for x in terms of y, which is what the question is asking for.

  x = log(y/4)/log(1.0256) . . . . . . . exact expression

When y=12, this is ...

  x = log(12/4)/log(1.0256) ≈ 43.46 . . . . weeks

_____

This is a linear equation in log(y), so can be written as such:

  x = 91.0912·log(y) -54.8424 . . . . . approximate expression