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### Population of China

```
Date: 05/13/97 at 01:12:40
From: Julie
Subject: continuous growth

I tried figuring this problem out and I think I got the answer, but I
don't know how I got it. Can you please help me and tell me if I am
right?  Can you also explain how to get the answer?  Here's the
problem:

Given birth rates and death rates, the population of China is assumed
to be growing continuously. If the population growth rate is 4.3
percent and the current population of the country is 1.27 billion
people, what year will the population reach 2 billion?

I got year 2008 and I don't know how.  I'd really appreciate it if you
could help me.

Thanks.
```

```
Date: 05/13/97 at 05:49:53
From: Doctor Mitteldorf
Subject: Re: continuous growth

Dear Julie,

You can solve this problem by "brute force" by multiplying 1.27 times
1.043, and continuing to multiply the resulting numbers by 1.043 as
many times as it takes to get to 2. You find that the number passes 2
after 11 multiplications, so 1997+11 = 2008 is the right answer. Is
this what you did?

There's a faster way to do this problem, but you need to know about
logarithms.  Take the log of 2 and subtract the log of 1.27.

log(2) = .3010
log(1.27) = .1038
log(2) - log(1.27) = .1972

This tells you that the log has to increase by .1972 as the population
increases.  But each year, the log of the population increases by:

log(1.043) = 0.01828

Now you can just divide .1972 by 0.01828 to get 10.8, so the
population will pass 2 billion in 10.8 years.

This logarithm method works because multiplying two numbers

-Doctor Mitteldorf,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Logs

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