# Thread: Finding Intersections of Polynomial and Exponential Functions

1. ## Finding Intersections of Polynomial and Exponential Functions

Hi. I'm looking for some general tips for solving equations that I'm running into as I study for the Math GRE Subject test. For example, I have a problem where I have to find the intersection of $\displaystyle y=x^{12}$ and $\displaystyle y=2^{x}$, and another where I need to find the intersection of $\displaystyle y=10x$ and $\displaystyle y=e^{x}$. Usually, one function is some sort of polynomial and the other is an exponential function. I don't even know what subject this falls under, so I'm not sure what sorts of textbooks to look into. Often, the questions just ask how many points of intersection there are, as opposed to what the values are, so maybe there is some sort of method for that that I don't know about.

Thanks.

2. Originally Posted by StarKid
Hi. I'm looking for some general tips for solving equations that I'm running into as I study for the Math GRE Subject test. For example, I have a problem where I have to find the intersection of $\displaystyle y=x^{12}$ and $\displaystyle y=2^{x}$, and another where I need to find the intersection of $\displaystyle y=10x$ and $\displaystyle y=e^{x}$. Usually, one function is some sort of polynomial and the other is an exponential function. I don't even know what subject this falls under, so I'm not sure what sorts of textbooks to look into. Often, the questions just ask how many points of intersection there are, as opposed to what the values are, so maybe there is some sort of method for that that I don't know about.

Thanks.
Can't be done with algebra (unless you have learned about the Lambert W-function, which I doubt you have). This sorts of questions are expecetd to be solved to a specified degree of accuracy (eg. 4 dp) using a graphics calculator or CAS technology. The source of the questions should make that clear.

3. How about some way to narrow it down to how many solutions there are?

4. Originally Posted by StarKid
How about some way to narrow it down to how many solutions there are?
Does the source of the question suggest using a calculator?

5. Originally Posted by mr fantastic
Does the source of the question suggest using a calculator?
No. It's a standardized test; I can't use anything but my brain.

6. Originally Posted by StarKid
No. It's a standardized test; I can't use anything but my brain.
Tough break.

For number of solutions, I suggest drawing sketch graphs of y = x^12 and y = 2^x and counting the number of intersection points. Ditto for the others.