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The population of a city increases by four percent every year.
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How many years does it take for the population of the city to double?
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This problem is an exponential growth problem.
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Weโve our starting population.
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And we need to multiply that by ๐ to the rate times time power.
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And that will give us ๐ด, our new amount.
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We donโt know the population of the city.
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But we do know weโre interested in it doubling.
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We can actually use the number one as the value of our population.
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If one is the starting value of our population, then two would be the doubled value.
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Weโre gonna multiply that by ๐ to the 0.04 power.
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0.04 is four percent written as a decimal.
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And our missing value is ๐ก.
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We want to know how many years it would take to go from one to two in our population.
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And that means we need to isolate ๐ก.
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We need to get ๐ก by itself.
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Anything multiplied by one is itself.
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To isolate ๐ก, we need to get it out of the exponent.
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And to do that, weโll need to take the natural log of ๐ to the 0.04๐ก power.
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And if we take the natural log on the right side of the equation, we need to take the natural log on the left side of the equation.
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The natural log of two will be equal to 0.04 times ๐ก.
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This is because the natural log of ๐ to any power equals whatever is in the exponent.
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Now that we know that and our goal is to isolate ๐ก, we can divide both sides of the equation by the rate 0.04.
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And ๐ก equals 17.328 continuing.
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So itโs just over 17 years, 17.3 years.
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But in this kind of question, we canโt put 17.3 years.
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We need to round to the nearest year.
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But if we rounded down to 17, the population would it be doubled all the way?
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Year 18 would be the first year that the population has doubled.
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So we say that it takes 18 full years for the population of the city to double.