# Math Help - Products and quotients of Functions

1. ## Products and quotients of Functions

A s,all town initially has a pop. of 1500. The population of people grows as a function of time: P(t) = 1500(1.5)^t

The surrounding farms supply the food. The amount of food, F, decreases as the size of farms decrease: F(t) = 5000(0.25)^t

Determine the point of intersection.
I understand that the intersection is when P(t) = F(t). We solve for t, then sub that value back into one of the original equations to get the value for coordinate (x,y)

I'm note sure how my teacher did that with the calculations, though.
From line 5 onward, got confused:

On line 6, he distributed the log and separated log 0.3 and tlog 1.5 with a "plus" symbol?
How is that possible?

When he gathered all the logs with the "t" in it to the right side, how did he isolate "t"?

Continuing with a similar question, what is the t-intercept of function y = F(t) - P(t)?
I found that:
y = F(t) - P(t)
y = 5000(0.25)^t - 1500(1.5)^t

I was stuck here because of the above.

2. Looks good. You can combine all the logs if you want too.

3. Originally Posted by TN17
A s,all town initially has a pop. of 1500. The population of people grows as a function of time: P(t) = 1500(1.5)^t

The surrounding farms supply the food. The amount of food, F, decreases as the size of farms decrease: F(t) = 5000(0.25)^t

Determine the point of intersection.
I understand that the intersection is when P(t) = F(t). We solve for t, then sub that value back into one of the original equations to get the value for coordinate (x,y)

I'm note sure how my teacher did that with the calculations, though.
From line 5 onward, got confused:

On line 6, he distributed the log and separated log 0.3 and tlog 1.5 with a "plus" symbol?
How is that possible?

When he gathered all the logs with the "t" in it to the right side, how did he isolate "t"?

(typo on line 2)

Line 5 uses the log property

$log\left(a^b\right)=blog(a)$

Line 6 uses the log property

$log(ab)=log(a)+log(b)$

"t" was isolated by "factoring"

$log0.3=t(log0.25-log1.5)$

Then divide both sides by the contents of the brackets.

4. Thanks a lot!