How to solve this equation using logs How do solve this equation for x using logarithms? 4^x = 6^x-3 If it is not possible using logarithms, please provide another way. Thank you in advance

Cristofer Watson

Cristofer Watson

Answered question

2022-10-22

How to solve this equation using logs
How do solve this equation for x using logarithms?
4 x = 6 x 3
If it is not possible using logarithms, please provide another way. Thank you in advance

Answer & Explanation

silenthunter440

silenthunter440

Beginner2022-10-23Added 19 answers

In the most general case, equations such as a x = b x + c do not show explicit solutions and numerical methods (such as Newton) must be used. Considering that we look for the zero of function
f ( x ) = a x b x c
starting from a reasonable guess x 0 , Newton method will update the guess according to
x n + 1 = x n f ( x n ) f ( x n )
Let us consider the case of your equation
f ( x ) = 6 x 4 x 3
f ( x ) = 6 x log ( 6 ) 4 x log ( 4 )
By inspection, f ( 1 ) = 1, f ( 2 ) = + 17 show that the solution is quite close to 1; so, let us use x 0 = 1 and apply the method. This will generate the following iterates
x 1 = 1.192109
x 2 = 1.161257
x 3 = 1.160180
x 4 = 1.160179
which is the solution for seven significant figures.
However, we can do it faster considering instead
g ( x ) = log ( 6 x ) log ( 4 x + 3 ) = x log ( 6 ) log ( 4 x + 3 )
which is much more linear. Doing the same would lead to the following iterates
x 1 = 1.154214
x 2 = 1.160170
x 3 = 1.160179

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