X=log1218 and Y=log2454. Find XY+5(X−Y)

Harold Kessler

Harold Kessler

Answered

2022-01-01

X=log1218 and Y=log2454. Find XY+5(XY)

Answer & Explanation

Andrew Reyes

Andrew Reyes

Expert

2022-01-02Added 24 answers

Let I=log18log12log54log24+5(log18log12log54log24) . Also, let log3=x and log2=y.
Then,
I=log322log223log332log233+5(log322log223log332log233)=2x+y2y+x3x+y3y+x+5(2x+y2y+x3x+y3y+x)
=6x2+5xy+y2+10x2+35xy+15y235xy10y2(2y+x)(3y+x)
=x2+5xy+6y2x2+5xy+6y2=1
Daniel Cormack

Daniel Cormack

Expert

2022-01-03Added 34 answers

Note that XY+5(XY)=(X5)(Y+5)+25, so it suffices to find (X5)(Y+5).
(X5)=log12(18)5=log1218125=log123329=3log12(24).
(Y+5)=log24(54)+5=log24(54245)=log24(21638)=8log24(12).
Multiplying together gives 24log12(24)log24(12)=24log12(12)=24.
Adding 25 to this gives 1, which is your answer.

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