Solve logarithmic equation 3^(log_3^2x) + x^(log_3x)=162

Rihanna Bentley

Rihanna Bentley

Answered question

2022-11-06

Solve logarithmic equation 3 log 3 2 x + x log 3 x = 162
Find x from logarithmic equation
3 log 3 2 x + x log 3 x = 162
I tried solving this, with basic logarithmic laws, changing base, etc., but with no result, then I went to wolframalpha and it says that its alternate form is:
2 e log 2 x log 3 = 162
But I don't know how it came to this result, can you help me guys?

Answer & Explanation

Sean Sutton

Sean Sutton

Beginner2022-11-07Added 17 answers

The following is how WolframAlpha simplified it:
162 = 3 log 3 2 x + x log 3 x = 3 log 3 x log 3 x + x log 3 x = ( 3 log 3 x ) log 3 x + x log 3 x = x log 3 x + x log 3 x = 2 x log 3 x = 2 ( e log x ) log 3 x = 2 e log x log 3 x = 2 e log x log x log 3 = 2 e log 2 x log 3
However, this is probably not the optimal process if you want to solve the problem; leaving things in terms of log 3 x is helpful.
Hallie Stanton

Hallie Stanton

Beginner2022-11-08Added 5 answers

3 log 3 2 x = ( 3 log 3 x ) log 3 x = x log 3 x

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