Expert Answer to "Solve the equation 112(log5(x))2−12⋅11(log5(x))2+11=0"

Jasper Dougherty

Answered question

2022-03-20

Solve the equation

11^{2 {(\log_{5} (x))}^{2}} - 12 \cdot 11^{{(\log_{5} (x))}^{2}} + 11 = 0

Ashton Conrad

Beginner2022-03-21Added 11 answers

Step 1
Let,

x > 0

then
Substitute

t = 11^{{(\log_{5} (x))}^{2}}

you get

t^{2} - 12 t + 11 = 0

(t - 1) (t - 11) = 0 .

which gives

t = 1, t = 11

Finally, you can solve the equation for new variable t:

11^{{(\log_{5} (x))}^{2}} = t

{(\log_{5} (x))}^{2} = \log_{11} t

\log_{5} (x) = \pm \log_{11} t

Step 2
Small Supplement:
The meat of this trick is as follows:

a^{b c} = a^{c b} = {(a^{c})}^{b} = {(a^{b})}^{c} .

pastuh7vka

Beginner2022-03-22Added 13 answers

Step 1
Since

{(a^{b})}^{c} = a^{b c}

{(11^{{(\log_{5} (x))}^{2}})}^{2} = 11^{2 {(\log_{5} (x))}^{2}}

so letting

u = 11^{{(\log_{5} (x))}^{2}}

, you get

u^{2} - 12 u + 11 = 0

which factors as

(u - 11) (u - 1) = 0

Now since

11 = 11^{1}

and

1 = 11^{0}

, you have three solutions where

\log_{5} (x) = 0, \pm 1

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