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If a number increases by 4% then you multiply the number by 1.04 (100%+4%) to find the new number. For repeated increases of 4% you multiply by 1.04 repeatedly. This leads to an equation for the population of Townsville:
where t is the number of years and P(t) is the population after that many years.
For % decreases we multiply by 100% minus the percent decrease. For Bayville's 2% decrease this would be 100%-2% = 98% = 0.98. This makes the population equation for Bayville:
The problem is to find the time when these populations are equal:
To solve an equation like this, with the variable in one ore more exponents, we will use logarithms. The base of the logarithm we use doesn't make any significant difference. Using base 1.04 or base 0.98 logarithms will lead to the simplest possible expression for the exact answer. But if you want a decimal approximation for the answer then choosing a base that your calculator "knows" (like base 10 or base e (aka ln)) would make finding the decimal easier. Since I assume you want a decimal number, I am going to use base e:
Next we will use a property of logarithms,
, to separate the initial population numbers from the exponential:
Next we will use another property of logarithms,
, to move the exponents of the argument out in front. (This property is the very reason we use logarithms on equations like this. It allows us to "extract" an exponent and put it "where we can get at it" with regular Algebra.)
ln(60000) + t*ln(1.04) = ln(150000) + t*ln(0.98)
Now that t is out of the exponent, we can solve for it. First we gather the t terms on one side and the other terms on the other side of the equation. Subtracting t*ln(0.98) and ln(60000) from each side we get:
t*ln(1.04) - t*ln(0.98) = ln(150000) - ln(60000)
Then we factor out t:
t*(ln(1.04) - ln(0.98)) = ln(150000) - ln(60000)
and divide both sides by (ln(1.04) - ln(0.98)):
This is an exact expression for the number of years it will take for the populations fot he two towns to be equal. For a decimal approximation we get out our calculators and find the four logarithms:
t = 15.4196901594447808
So it will take almost 15 and a half years for the populations of the two towns to be equal.