We are in a log unit and I have no idea how to do this

Question

The population P of a city is given by P=2500e^kt, where t is the time in years, with t=0 corresponding to the population in 1950. In 1995, the population was 1,400 people.  Find the value of k and use it to estimate the number of people in the year 2010.

Answer 1

Hi, Lydia!

 

This question is very common in a log unit.  The basic outline for solving this question is to use the information they gave you (year = 1995, pop = 1400) to find k.  Once you know k, find t for the year 2010 and plug everything in to find the population in 2010.

 

Part A. Finding k:

year 1995 --> t=45  (1995-1950 = 45)

P = 1,400

 

P = 2500e^kt

1400 = 2500e^k*45

1400/2500 = e^45k

0.56 = e^45k

 

We need to somehow get k out of the exponent in order to be able to solve for it.  Logarithms are the "opposites" to situations that have exponents in the same way that multiplication is the opposite of division and square rooting is the opposite of squaring.  We can use a log (or, more specifically, a log with a base of e called a natural log or "ln") to get 45k out of the exponent.

 

Start by taking the ln of both sides.  Reminder ln(a) = log_e(a).

ln(0.56) = ln(e^45k)

ln(0.56) = 45k*ln(e) <--This uses the exponential logarithmic rule that states that log(a^b) = b*log(a).

 

ln(e) = 1, so let's substitute that back in:

ln(0.56) = 45k

ln(0.56)/45 = k

k = ln(0.56)/45 = -0.0129

 

Part B. Finding the population in 2010:

t = 60 (2010-1950 = 60)

k = -0.0129 (from Part A)

 

P = 2500e^-0.0129*60

P = 2500e^-.774

P = 1152.9 --> approximately 1153 people

 

I hope this was helpful!

 

Patty S.

Answer 2

For every growth or decay problem, if they don't provide you with the k constant value, you have to calculate what it is with the given information.

 

I always underline the important information given.

 

"The population P of a city is given by P=2500e^kt, where t is the time in years, with t=0 corresponding to the population in 1950. In 1995, the population was 1,400 people. Find the value of k and use it to estimate the number of people in the year 2010."

 

So we see that they want you to find the value of k.

 

P is the population after a certain amount of time, t.

 

They tell you that in 1995, the population was 1400 people, so P = 1400 and t is the amount of years between 1950, where t is 0 and 1995, so t = 45.

 

Now we plug the values into the formula:

 

1400 = 2500 * e ^ (k * 45)

 

In order to solve for k, we must isolate it so first we want to divide by 2500, both sides.  It cancels out the 2500 on the right side, leaving us with:

 

1400/2500 = e ^ (k * 45)

 

Now we have k as an exponent and we have to bring it out of the exponent, and the only way to do that is by using our Logarithm Properties, specifically the Exponent Rule.  We can only use that rule if we have a log or ln, and since we don't, we must introduce it ourselves and as always, if we do it to one side, we must do it to the other.  Since we are working with e, we must use ln

 

ln(1400/2500) = ln(e)^(k*45)

 

Using our Exponent Rule, we can bring any exponent of a log down to the coefficient of that same long, giving us the following:

 

ln(1400/2500) = (k*45)ln(e)

 

Now since ln(e) = 1, we have the following:

 

ln(1400/2500) = (k*45)

 

To solve for k, we divide both sides by 45, and you punch all that in your calculator giving you the value of k.

 

ln(1400/2500) ÷ 45 = k.

 

You then use that value for k and figure out what the value for t will be in the year 2010.  With those two values, you plug them back into the original P=2500e^kt equation and solve for P.

 

I didn't want to do all the work for you, I think I have guided you in the right direction and you can do it on your own, :)