Solving \(\displaystyle{1}+{x}^{{{10}+{\log{{x}}}^{{{10}+{\log{{x}}}^{{{10}+\cdots}}}}}}={\frac{{{1}}}{{{\log{{x}}}}}}\) To solve this equation I used

svrstanojpkqx

svrstanojpkqx

Answered question

2022-03-31

Solving 1+x10+logx10+logx10+=1logx
To solve this equation I used t=1+x10+logx10+logx10+ . LHS is equal to,
1+t=1+x10+logt
t=x10+logt
logt=(logx)×(10+logt)
x=elogt10+logt
Hence the equation is equivalent to
t+1=10+logtlogt
tlogt=10
I'm not sure if this helps in solving the equation.

Answer & Explanation

Mikaela Winters

Mikaela Winters

Beginner2022-04-01Added 14 answers

You don't have to worry about the base, Let's just take the base to be b then
tlogb(t)=10
tln(t)=10ln(b)
W(ln(t)eln(t))=W(10ln(b))
ln(t)=W(10ln(b))
t=eW(10ln(b))
That makes,
x=eW(10ln(b))10+W(10ln(b))

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