Question

Solve 2x+1=2^x to find the other (not 0,1) intersection. Truly vexing. Thanks.

Answer 1

x=2.66

i plugged it in my graphing calculator

Answer 2

Yeah, I can get the same x value if I use a grapher s/w but I need to solve it mathematically. Well, "need" is a bit extreme. I just "want" to solve this purely for intellectual purposes. Thanks.

Answer 3

I noticed your post was basically left unanswered.

To solve your equation requires some rather sophisticated math.

I used "Newton's Method" to solve it.

Newton's Method can be summarized by :

newx = x - f(x)/f'(x)

for yours:
newx = x -(ln2(2^x) - 2x - 1)/(ln2(2^x)-2)

you would then start with a reasonable guess as an answer. I went with x=2

and my newx was 2.360674
now make that your x and sub again.

The x value you just used and the newx should approach each other.

after 11 steps of doing this on my calculator I had x = 2.62975

sub that back into your equation gives me
Left side = 6.3195
Right side = 6.31923 , not bad

If you need more accuracy, just keep repeating the process until your calculator shows the same output as input, ( x = newx)

Newton's Method has been known for hundrreds of years, (Isaac Newton 1643-1727)
but only since the advent of scientific calculators has it become really practical.

Answer 4

Wow! Cheers. I'll look up on it. But, basically, there's no way to simplify that "equation" to something like x=... ? I was thinking along the lines of using Log, but I can't simplify log(2x-1) :(
Again, cheers.