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.

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