solve for x without using softwares log_(sqrt(x))2+log_6x^x=4

metodystap9

metodystap9

Open question

2022-08-19

solve for x without using softwares log x 2 + log 6 x x = 4
Is there any nice way to solve this equation without wolfram?
log x 2 + log 6 x x = 4
Thanks.

Answer & Explanation

Adelyn Mercado

Adelyn Mercado

Beginner2022-08-20Added 13 answers

Rewriting
log x 2 + log 6 x x = 4
leads to the equation
f ( x ) = x log 2 ( x ) 4 log ( 6 ) log ( x ) + log ( 4 ) log ( 6 ) = 0
which cannot be solved analytically ( f ( x ) = 0 or f ( x ) = 0 cannot be solved either).
Plotting the function shows two roots, the first close to x = 1.5, the second close to x = 4.0. For the solution, Newton is the simplest root-finder. Strating from a "reasonable" guess x 0 , it will update it according to
x n + 1 = x n f ( x n ) f ( x n )
For the first root, using x 0 = 1.5, the successive iterates are 1.45386, 1.45521 which is the solution for six significant figures. Limiting to a single iteration, we should have obtained
x 6 log ( 9 8 ) log ( 6 ) 3 log ( 384 ) 3 log 2 ( 3 2 ) log ( 147456 )
For the second root, using x 0 = 4.0, the successive iterates are 3.91886, 3.91689 which is the solution for six significant figures. Limiting to a single iteration, we should have obtained
x log ( 4096 81 ) + log ( 6 ) log ( 64 ) log 2 ( 4 ) + log ( 8 3 )
As you can see, all of this could be done using a pocket calculator.
Taliyah Reyes

Taliyah Reyes

Beginner2022-08-21Added 6 answers

I am pretty sure the isn't solvable with Algebra II, unless you resort to the mess that is complex logs. However:
l o g x 2 + l o g 6 x x = 4 l o g x 6 + x ( l o g 2 6 ) ( l o g 6 x ) = 4 ( l o g 2 6 ) 2 l o g 6 x + x ( l o g 2 6 ) ( l o g 6 x ) = 4 ( l o g 2 6 ) 2 + x ( l o g 6 x ) 2 4 ( l o g 6 x ) ( l o g 2 6 ) = 0
which is a quadratic in l o g 6 x except for the pesky x coefficient. Using the quadratic formula anyway, and evaluating the logs,
l o g 6 x 10.34 ± 106.91 8 x 2 x
This is definitely solvable. However, The only way I can come up with a numerical solution is graphing the two sides.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?