How to solve the equation x^2(log_(10)x)^5=100? Solving few steps I got x=10^((x/(10))^5).

miklintisyt

miklintisyt

Answered question

2022-10-26

How to solve the equation x 2 ( log 10 x ) 5 = 100?
Find the value of x (without hit and trial) from the equation x 2 ( log 10 x ) 5 = 100
Solving few steps I got x = 10 ( x 10 ) 5

Answer & Explanation

Davin Meyer

Davin Meyer

Beginner2022-10-27Added 13 answers

l o g 10 x = y 10 y = x x 2 = 10 2 y
The equation will then become
10 2 y y 5 = 100
One solution that I see is y = 1, which will give x = 10. Why do you wish to solve this equation without hit-and-trial method?
Chaim Ferguson

Chaim Ferguson

Beginner2022-10-28Added 4 answers

Just for fun. The solution of
x 2 ( log 10 ( x ) ) 5 = M
in terms of the Lambert W function:
x = 4 10 M ( ln 10 ) 5 / 2 125 W ( 2 5 ( ln 10 ) M 1 / 5 ) 5 / 2
to get all solutions, use all branches of W. Numerically, with M = 10
0.0031090972245807637012 0.0057756764312723178989 i 0.0085157013468260475802 0.018595683863231787939 i 0.070950674893432382960 0.16326665000276959405 i 10.00000000000000000 0.070950674893432382960 + 0.16326665000276959405 i 0.0085157013468260475802 + 0.018595683863231787939 i 0.0031090972245807637012 + 0.0057756764312723178989 i

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