Solving logarithmic equations including x Let log_3(x−2)=6−x It's obvious drawing the graphs of the two functions that the only solution is x=5. But this is not really a proof, rather than observation.

Hallie Stanton

Hallie Stanton

Answered question

2022-11-06

Solving logarithmic equations including x
Let
log 3 ( x 2 ) = 6 x
It's obvious drawing the graphs of the two functions that the only solution is x = 5. But this is not really a proof, rather than observation.
How do you prove it algebraically?

Answer & Explanation

Claudia Woods

Claudia Woods

Beginner2022-11-07Added 15 answers

We can solve this equation for x in closed form as follows.
Write log 3 ( x 2 ) = 6 x. Then we have
(1) x 2 = e log 3 [ 4 ( x 2 ) ] = 3 4 e ( log 3 ) ( x 2 )
Thus, multiplying ( 1 ) by log 3 and rearranging terms gives
( log 3 ) ( x 2 ) e ( log 3 ) ( x 2 ) = 3 4 log 3
Recalling that Lambert's W function is defined as z = W ( z ) e W ( z ) gives
x = 2 + W ( 3 4 log 3 ) log 3 = 2 + 3 = 5
as was to be shown!
Owen Mathis

Owen Mathis

Beginner2022-11-08Added 3 answers

Since one has x + log 3 ( x 2 ) = 6, let f ( x ) = x + log 3 ( x 2 ). Note that x > 2.
Now,
f ( x ) = 1 + 1 ( x 2 ) ln 3 > 0.
Hence, since y = f ( x ) is increasing for x > 2, we know that f ( x ) = 6 has at most one real solution (you know that it is x = 5).

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