How do you solve for x in x+log_2x=6 ?

2 Answers
May 13, 2018

x=4

Explanation:

This is a transcendental equation since x is in the logarithm and not. Therefore, it does not in general have an easy solution and there isn't any perfect method for solving it.

This one does have a simple solution, but this is because this function had to be selected very carefully.

Let's assume there's some integer n such that 2^n = x. The equation then becomes
2^n + n = 6

Plugging in values of n,
n = 0 implies 2^n + n = 1 ne 6
n = 1 implies 2^n + n = 2 ne 6
n = 2 implies 2^n + n = 6 = 6

So the solution x=4 happens to be the right solution.

May 13, 2018

One cannot solve the equation x+log_2x=6 using algebraic methods.

Explanation:

One can use a graphical method by converting the base to base e and then separating into two equations.

Convert to base e:

x+ln(x)/ln(2)=6

Subtract x from both sides:

ln(x)/ln(2)=6-x

Multiply both sides by ln(2)

ln(x)=6ln(2)-xln(2)

Separate into two equations:

color(red)(y = ln(x)) and color(blue)(y= 6ln(2)-xln(2))

Graph the two equations:

www.desmos.com/calculatorwww.desmos.com/calculator

Please observe that the two graphs intersect at x = 4

You can use a recursive computation method such as Newton's Method to approximate the solution. Because the method requires many lines of computation, I will not do it, here, but I have provided a link so that you may read about it.

The easiest way to obtain a solution is to enter the original equation into WolframAlpha

Please open the above link and observe that WolframAlpha has computed the solution and it is x = 4.