Solve 2x+1=2^x to find the other (not 0,1) intersection. Truly vexing. Thanks.
x=2.66
i plugged it in my graphing calculator
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.
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.
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.
To solve the equation 2x + 1 = 2^x, we need to find the value of x for which the equation is true, other than x = 0 or x = 1.
Unfortunately, this equation cannot be solved algebraically using conventional methods. However, we can find an approximate solution using numerical methods.
One common numerical method is the "trial and error" method, where we substitute different values of x into the equation until we find a value that satisfies it. To make this process easier, we can use a graphing calculator or software to plot the two functions, y = 2x + 1 and y = 2^x, and find their point of intersection visually.
Here's a step-by-step process to find the approximate solution:
1. Graph the functions: y = 2x + 1 and y = 2^x on the same set of axes.
2. Observe the graph and identify the point(s) of intersection between the two curves. We are looking for a point other than x = 0 or x = 1.
3. Zoom in or adjust the window of your graphing calculator/software to get a closer look at the region where the two graphs intersect.
4. Use the zoomed-in view to estimate the x-coordinate of the intersection point.
5. Once you have estimated a value for x, substitute it back into the original equation 2x + 1 = 2^x to check if it satisfies the equation.
6. If the value satisfies the equation, you have found the approximate solution. If not, adjust your estimate and repeat steps 4 to 6 until you find a value that satisfies the equation.
Keep in mind that this method provides an approximate solution and may not give an exact value. It's also important to be cautious when relying solely on numerical methods, as they do not guarantee the existence of all possible solutions.