Question

Part A: Explain why the x coordinates of the points where the graphs of the equations y=4^x and y=2^x+2 interest are the solutions of the equation 4^x=2^x+2

Answer 1

Transitive

If A=B and B=C, then A=C

Answer 2

I don't get it

Answer 3

Let's say we have two equations.

\[y = 2 x\]and\[y = -2x\]
If we want to find where their intersect on a graph, we need to understand how the graph relates to the equation. Remember how we graph simple equations. We let x = 1 then find y and then plot the point. Repeat for a few more values. The graph represents the value of y for each value of x of a given equation.

If two equations intersect, that means that both equations have the same y-value given the same x-value.

That means that we can set the two equations equal to each other, because they have the same y-value. From my earlier two equations, \[2x = - 2x\]

Now, we can solve for the x-value that satisfies this condition of both equations having the same y-value.

In this case, it is zero. Both graphs intersect at zero.

Answer 4

Both graphs intersect at (0,0) to be more precise.

Answer 5

@nikato is he correct?

Answer 6

Yes. Same process for solving your problem, but he was using his own example

Answer 7

how did he find x?

Answer 8

how did he find x though?

Answer 9

My example was a bad one. You divide both sides by x and then are left with 2 = -2. This is arbitrary.

If we subtract -2x from both sides, we get 2x + 2x = 0. The only value of x that satisfies the equation is zero.

4x = 0 --> x = 0

because 0/4 = 0.

A better example might be \[y= x^2\]and\[y = 2x+3\]

We set them equal to each other. \[x^3 = 2x+3\]

Set the equation equal to zero\[x^2 - 2x- 3 = 0\]

Now it looks like your run-of-the mill quadratic equation.

Answer 10

You'll get two solutions because.

Answer 11

Can you explain with my equation?

Answer 12

hang on let me try one moment please.

Answer 13

Sure.

\[y = 4^x\]and \[y = 2^x + 2\]

Answer 14

Without spoiling your work, here is a neat visualization.

http://www.wolframalpha.com/input/?i=y+%3D+4%5Ex+and+y+%3D+2%5Ex+%2B+2

Answer 15

To find out the x coordinate you will do 4^x=2^x+2 and solve for x by subtracting 2^x from both sides leaving you with 2^x=2. Then, by subtracting 2 from both sides leaving you with 2^x-2=0. what do I do next?

Answer 16

You cannot subtract 4^x and 2^x because they are dissimilar bases.

Answer 17

can you explain

Answer 18

Getting these to the same base is easy in this case.

Remember that\[2^2 = 4\]

Therefore, we can rewrite the equation as\[4^x = (2^2)^x\]Recall that\[(x^y)^z = x^{yz}\]

Therefore, \[2^{2x} = 2^x + 2\]

Now they're in the same base.

Answer 19

Solve as you previously attempted.

Answer 20

so 2^2x-2^x-2=0?

Answer 21

Yep!