1. ## Exponential Functions

I need the steps to this problem explained, please.

In the year 2000, the population of Virginia was about 7,400,000. Between the years 2000 and 2004, the population in Virginia grew at a rate of 5.4%. At this growth rate, the function f(x) = 7,400,000 (1.054)^x gives the population xyears after 2000.
In what year will the population reach 15,000,000?
In what year will the population reach 20,000,000?

Thanks.

2. f(x) represents the population after x amount of years. So, you are given f(x) = 15 000 000 and 20 000 000 and must solve for x (i.e. the amount of years after 2000). So:

$15 000 000 = 7 400 000 (1.054)^{x}$
$\frac{150}{74} = (1.054)^{x}$
$\frac{75}{37} = (1.054)^{x}$

Take the logarithms of both sides and solve for x. Come back if you still have trouble!

Edited: Misread 7 400 000 for 74 000 000

3. Originally Posted by o_O
f(x) represents the population after x amount of years. So, you are given f(x) = 15 000 000 and 20 000 000 and must solve for x (i.e. the amount of years after 2000). So:

$15 000 000 = 74 000 000 (1.054)^{x}$
$\frac{15}{74} = (1.054)^{x}$

Take the logarithms of both sides and solve for x. Come back if you still have trouble!
I'm sorry, what are logarithms?

4. Have you ever seen this notation before?
$\log(x) \quad \text{or} \quad \ln (x)$

5. Originally Posted by o_O
Have you ever seen this notation before?
$\log(x) \quad \text{or} \quad \ln (x)$
No...

6. Well this involves playing with some numbers then. For example, if you plug in x = 13 (i.e. the year 2013) you find that the population is 14 660 867 (just a bit less than 15 000 000) while when you plug in x = 14 you find that the population is 15 454 554 (just a bit over). So, you can conclude that somewhere in the year 2013, the population will reach 15 000 000.