How to solve this logarithm equation log_2 (log_3 (log_4(x^(3x))))=0

Aleseelomnl

Aleseelomnl

Answered question

2022-08-13

How to solve this logarithm equation
log 2 { log 3 [ log 4 ( x 3 x ) ] } = 0
How would I go about solving this? I tried doing log 4 ( x 3 x ) ) = 0 but I don't know how to incorporate the other logs

Answer & Explanation

vladinognm

vladinognm

Beginner2022-08-14Added 13 answers

If log 2 ( soemthing ) = 0, then something = 1
So log 3 log 4 ( x 3 x ) = 1
If log 3 ( something ) = 1, then something = 3
So log 4 ( x 3 x ) = 3.
Now recall that log 4 ( x 3 x ) = 3 x log 4 x, so we get 3 x log 4 x = 3
It follows that x log 4 x = 1. That implies log 4 ( x x ) = 1, so x x = 4
Doubtless you know that 2 2 = 4, so x = 2 is a solution. If x > 2 then x x > 4. If x < 2 then x x < 4, although that last fact is a bit more involved than it might superficially look.
kaeisky9u

kaeisky9u

Beginner2022-08-15Added 4 answers

Work from the outside in. Recall that:
log b ( y ) = a b a = y
Hence, we get:
2 0 = log 3 ( log 4 ( x 3 x ) ) 3 1 = log 4 ( x 3 x ) 4 3 = x 3 x
Now notice that 4 3 = ( 2 2 ) 3 = 2 6 . By inspection, we see that x = 2 is one possible solution.

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