Question

HELP PLEASE!!!
The population (in millions) of a certain country can be approximated by the function P(x)=50(1.03)^x, where x is the number of years since 2000. In which year will the population reach 100 million? Hint: An answer such as 2002.4 would represent the year 2002.

Answer 1

can I just guess and check for "x" untill I get 100,000,000?

Answer 2

You could just guess and check. But there is a way for solving equations with the variable being a exponent. It requires the use of logarithms. You have \(P(x)=50(1.03)^x\) and you know that \(P(x)=100\).
So you have the equation
\[100=50(1.03)^x\]
You need to solve for \(x\). So the process is:
\[\eqalign{
&100=50(1.03)^x \\
&2=(1.03)^x \\
&log_{10}(2)=x\phantom{.}log_{10}(1.03) \\
& \\
&\frac{log_{10}(2)}{log_{10}(1.03)}=x \\
&x\phantom{.}\dot{=}\phantom{.}23.45\phantom{.}\dot{=}\phantom{.}23
}\]

Answer 3

To prove it:
\[P(23.45)=50(1.03)^{23.45}\dot{=}50(2.0000134641)\dot{=}100\]

Answer 4

I realize now why I was so confused because I forgot to appropriately put the 100 million in decimal form, thank you Keith!

Answer 5

Anytime chowbelly!