APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

(Population Word Problems)

To solve an exponential or logarithmic word problem, convert the narrative to an equation and solve the equation.

Now we are going to get serious.

Example 18: Iraq has one of the largest population growth rates in the world, and United Kingdom has one of the lowest. In 1990 the population of Iraq was 18 million and the population of United Kingdom was 57 million. The relative growth rates are 3.4% and 0.3%, respectively. The relative growth rates are based on a base of e. When, if ever, will the populations of both countries be equal?

Solution and Explanations:

Let the start of our study take place in the year 1990 (t = 0 in 1990). The population equation for Iraq is

and the population equation for United Kingdom is

Recall, we were given the relative growth rates and starting populations.

• Since we want to know when f(t) will equal k(t), set the equations equal to each other and solve them for t.

  

• Divide both sides by 18,000,000 and by .

  

• Simplify the left side of the equation:

  

• Take the natural logarithm of both sides of the equation .

  

• Simplify the left side of the equation using the third rule of logarithms:

  

• Divide both sides of the equation by 0.031.

  

• The populations will be equal in the year 1990 + 37.183 = 2027.183 or 2027.

Note that this prediction is really out in space. Too many things will happen during the 37 years to disrupt the models. Just remember that when you calculate the answer to a problem similar to the above, it is really in its most simplistic form. At this point in your studies, you do not have the expertise to work with all the variables at work in population studies.

If you would like to work another example, click on Example

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