x^4 = 4^x ??? Is that what you mean?
4 ln x = x ln 4
ln x / x = ln 4/4 = .347
I graphed it. When x = 2, lnx /2 = .347
but
it is pretty obvious that
2^4 = 16 = 4^2 = 16
How many roots does the graph of this function have?
4x^4 - 4^x
I believe that it either has two or three roots.
10 answers
No, I meant roots on a graph. More specifically, the number of times the graph passes through the x-axis (so, I suppose all cases where y=0). When I graphed the function on a graphing calculator, I believe I found three. There is a sort of parabolic part and then what appears to be a straight vertical line around x=6. I just wanted to check my answer.
Well, when the function is zero, then x^4 = 4^x and x = 2
There may be other roots, I will check.
There may be other roots, I will check.
I graphed it with
http://rechneronline.de/function-graphs/
and found one root at about -.6 and a second at one
http://rechneronline.de/function-graphs/
and found one root at about -.6 and a second at one
I should have had
4x^4 = 4^x
ln 4 x^4 = ln 4 + 4 ln x
so
ln 4 + 4 ln x = x ln 4
4x^4 = 4^x
ln 4 x^4 = ln 4 + 4 ln x
so
ln 4 + 4 ln x = x ln 4
Note when x = 1 you have
4* 1^4 - 4^1
which is
4 - 4
which is indeed zero
4* 1^4 - 4^1
which is
4 - 4
which is indeed zero
when you graph it use
4*x^4 - 4^x
in the box and hit enter
4*x^4 - 4^x
in the box and hit enter
Damon, I don't think that the graphing probram you linked to shows the correct graph
clearly as x ---> +large , 4^x > 4x^4, so the result becomes negative , the graph shows the function to be increasing and positive.
I tried Wolfram's graph
http://www.wolframalpha.com/input/?i=4x%5E4+-+4%5Ex
and it shows 3 x-intercepts
clearly at x = 1
another at about -.6
also at x = 6, the function is positive
at x=7 it is negative, so there is another between 6 and 7
clearly as x ---> +large , 4^x > 4x^4, so the result becomes negative , the graph shows the function to be increasing and positive.
I tried Wolfram's graph
http://www.wolframalpha.com/input/?i=4x%5E4+-+4%5Ex
and it shows 3 x-intercepts
clearly at x = 1
another at about -.6
also at x = 6, the function is positive
at x=7 it is negative, so there is another between 6 and 7
Yes, thanks !
Thank you Reiny, this is what I found when I graphed the function. The reason that these other graphs are not showing the third intercept is because they are zoomed in too closely to observe it.