# Applications of Exponential Equation Question

• Oct 26th 2010, 01:08 AM
BinaryBoy
Applications of Exponential Equation Question
Hi everyone,

Recently I have been working with applying exponential equations to the real world. However, one question completely threw me as is laid out below:

Q. The population of a town, N, is modeled by the function N = 15000(2 ^ 0.01t), where "t" is the number of years since 1980. In what year will the population reach 20000 people?

A. Okay, well to start off with, the general form of an exponential equation is:

y = ab ^ x

I know that "a" is the initial value and in this case is 15000.

N = 15000(2 ^ 0.01t)

I substitute N for 20000

20000 = 15000(2 ^ 0.01t)

Then do I divide both sides by 15000? And Once I have done that, and have transformed the equation into logarithmic form, how do I solve for "t"? Normally, it's quite easy, but in this case, the power of "t" is prefixed with "0.01", and I am unsure how to work with that.

Any help on solving this will be much appreciated!

Thanks,

Nathaniel.
• Oct 26th 2010, 02:16 AM
sa-ri-ga-ma
The given problem is in the form

$y = ab^{xt}$

Take log on both side

$ln(y/a) = xt*ln(b)$

Substitute the values and find t.
• Oct 26th 2010, 01:59 PM
HallsofIvy
Quote:

Originally Posted by BinaryBoy
Hi everyone,

Recently I have been working with applying exponential equations to the real world. However, one question completely threw me as is laid out below:

Q. The population of a town, N, is modeled by the function N = 15000(2 ^ 0.01t), where "t" is the number of years since 1980. In what year will the population reach 20000 people?

A. Okay, well to start off with, the general form of an exponential equation is:

y = ab ^ x

I know that "a" is the initial value and in this case is 15000.

N = 15000(2 ^ 0.01t)

I substitute N for 20000

20000 = 15000(2 ^ 0.01t)

Then do I divide both sides by 15000? And Once I have done that, and have transformed the equation into logarithmic form, how do I solve for "t"? Normally, it's quite easy, but in this case, the power of "t" is prefixed with "0.01", and I am unsure how to work with that.

Once you have solved for 0.01t, divide both sides by 0.01!

Quote:

Any help on solving this will be much appreciated!

Thanks,

Nathaniel.
• Oct 27th 2010, 02:36 AM
BinaryBoy
Thanks for the help so far, sa-ri-ga-ma and HallsofIvy. Much appreciated.

The answer should be 2022, but I am not getting that. Am I supposed to divide the log of 20000 by the log of 15000, or just log(20000 / 15000)?

And from there, do I then divide the result by the log of 2?
• Oct 31st 2010, 12:59 AM
sa-ri-ga-ma