Logarithm Equality sqrt(log_x(sqrt(3x))} * log_3 x = -1 I am not entirely sure how to go about solving for x. I cannot square each side because the product isn't >=0, I can't think of any more approaches right now.

Sara Fleming

Sara Fleming

Answered question

2022-09-23

Logarithm Equality
log x ( 3 x ) log 3 x = 1
I am not entirely sure how to go about solving for x. I cannot square each side because the product isn't 0, I can't think of any more approaches right now.

Answer & Explanation

Santiago Collier

Santiago Collier

Beginner2022-09-24Added 8 answers

Since square roots are nonnegative, we know that log 3 x < 0 so that 0 < x < 1. Now let k = log 3 and let y = log x. Then by using log rules, our equation becomes:
log 3 x log x log x log 3 = 1 1 2 log 3 x log x log x log 3 = 1 1 2 ( log 3 + log x ) log x = log 3 log x 1 2 ( k + y ) y = k y 1 2 ( k + y ) y = k 2 y 2 k y + y 2 = 2 k 2 y 2 + k y + 1 4 k 2 = 9 4 k 2 ( y + 1 2 k ) 2 = 9 4 k 2 y + 1 2 k = ± 3 2 k y = k , 2 k
Converting back, we find that either log x = log 3 or log x = 2 log 3 = log ( 1 / 9 ) so that either x = 3 or x = 1 / 9. But then since 0 < x < 1, we reject the first extraneous solution and conclude that x = 1 / 9
misyjny76

misyjny76

Beginner2022-09-25Added 1 answers

Will things look better without logarithms in the equation? First, let's simplify log x 3 x
log x 3 x = 1 2 log x 3 x = 1 2 ( log x 3 + log x x ) = 1 2 ( log x 3 + 1 )
Now, we'll use the info from Ciapan's comment: log x 3 = 1 log 3 x . The equation now becomes
1 2 ( y + 1 ) y = 1
Can you solve that? You may want to check your answers after to make sure they make sense.

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