Check the solution to "X=log1218 and Y=log2454. Find XY+5(X−Y)"

Harold Kessler

Answered question

2022-01-01

X = \log_{12} 18

and

Y = \log_{24} 54

. Find

X Y + 5 (X - Y)

Andrew Reyes

Beginner2022-01-02Added 24 answers

Let

I = \frac{\log 18}{\log 12} \cdot \frac{\log 54}{\log 24} + 5 (\frac{\log 18}{\log 12} - \frac{\log 54}{\log 24})

. Also, let

\log 3 = x

and

\log 2 = y .

Then,

I = \frac{{\log 3}^{2} \cdot 2}{{\log 2}^{2} \cdot 3} \cdot \frac{{\log 3}^{3} \cdot 2}{{\log 2}^{3} \cdot 3} + 5 (\frac{{\log 3}^{2} \cdot 2}{{\log 2}^{2} \cdot 3} - \frac{{\log 3}^{3} \cdot 2}{{\log 2}^{3} \cdot 3}) = \frac{2 x + y}{2 y + x} \cdot \frac{3 x + y}{3 y + x} + 5 (\frac{2 x + y}{2 y + x} - \frac{3 x + y}{3 y + x})

= \frac{6 x^{2} + 5 x y + y^{2} + 10 x^{2} + 35 x y + 15 y^{2} - 35 x y - 10 y^{2}}{(2 y + x) (3 y + x)}

= \frac{x^{2} + 5 x y + 6 y^{2}}{x^{2} + 5 x y + 6 y^{2}} = 1

Daniel Cormack

Beginner2022-01-03Added 34 answers

Note that

X Y + 5 (X - Y) = (X - 5) (Y + 5) + 25

, so it suffices to find

(X - 5) (Y + 5)

(X - 5) = \log_{12} (18) - 5 = \log_{12} \frac{18}{12^{5}} = \log_{12} 3^{- 3} 2^{- 9} = - 3 \log_{12} (24) .

(Y + 5) = \log_{24} (54) + 5 = \log_{24} (54 \cdot 24^{5}) = \log_{24} (2^{16} 3^{8}) = 8 \log_{24} (12) .

Multiplying together gives

- 24 \log_{12} (24) \log_{24} (12) = - 24 \log_{12} (12) = - 24 .

Adding

25

to this gives

1

, which is your answer.

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