Logarithmic AP

Calculus Level 3

If log ⁡ 3 2 , log ⁡ 3 ( 2 x − 5 ) a n d log ⁡ 3 ( 2 x − 7 2 ) \log _{ 3 }{ 2 } ,\log _{ 3 }{ ({ 2 }^{ x } } -5)\quad and\quad \log _{ 3 }{ ({ 2 }^{ x }-\frac { 7 }{ 2 } ) } are in AP, find x x


The answer is 3.

Tarun Singh by Tarun Singh , 23, India

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1 solution

Tarun Singh
Mar 20, 2015

2 l o g 3 ( 2 x − 5 ) = l o g 3 2 + l o g 3 ( 2 x − 7 2 ) 2 l o g 3 ( 2 x − 5 ) = l o g 3 ( 2 ∗ ( 2 x − 7 2 ) ) L e t , y = 2 x l o g 3 ( y − 5 ) 2 = l o g 3 ( 2 y − 7 ) y 2 + 25 − 10 y = 2 y − 7 y 2 − 12 y + 32 = 0 y = 8 , 4 2log_{3}{(2^x-5)} = log_{3}{2}+log_{3}{(2^x-\frac{7}{2})} \\ 2log_{3}{(2^x-5)} = log_{3}(2*(2^x-\frac{7}{2})) \\ Let, \quad y=2^x \\ log_{3}({y-5})^2 = log_{3}(2y-7)\\ y^2+25-10y=2y-7\\ y^2-12y + 32=0\\ y=8,4
Therefore, x = 3 x=3 or 2 2
As x = 2 x=2 will result in l o g 3 ( 4 − 5 ) log_{3}{(4-5)} = l o g 3 ( − 1 ) log_{3}{(-1)}
Therefore, x = 3 x=3


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