Question

The population of a city is decreasing by 4.5% each year. The population in the year 2010 started at 8,000 people. Make a table to represent how many years it would take to reach 3000 people.

Answer 1

Answer:

log(.375)/log(.955) = x

Step-by-step explanation:

To decrease by 4.5% means you multily by 1-.045 or .955.  It might be easier to see if you write it as 8000 - 8000 * .045.  Either way, now you can write it as 8000*.955.

Now, if this happened again you would multiply by .955 twice, or in other words have 8000 * .955^2.  You can keep going with this, which shows you can make this an exponential function.  8000*.955^x where x is the number of years the population decreases by 4.5%

The question then says how many times would we hae to decrease by 4.5% to get to 3000.  One way to do this is try every x value until you get there.  That usually takes a long time, so instead we use logs.

If you don't understand logs let me know, but basically it takes exponential functions and turns them into something else.  specificlaly it takes a = b^x where a and b are just numbers and x is the exponent, and turns it into log_b(a) = x or log base b of a equals x.  This will let us solve for x as a variable.  

To enter this into a calculator jut use log(a)/log(b) or ln(a)/ln(b).

Anyway, we start with 3000 = 8000*.955^x  That's not quite what we need to start with so divide both sides by 8000

3000/8000 = .955^x

3/8 = .955^x

.375 = .955^x

Then turn it into logarithmic form

log_.955(.375) = x

then just solve with the method I showed

log(.375)/log(.955) = x