Stream Of Powers

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After 2 Hours of solving math problems , john decided to have a break . he drank a cup of coffee to continue his work , while he was drinking the coffee he decided to create a problem and to solve it , but he deadly tired , he had a long day . although he drank a cup of coffee but he feels down and he wouldn't continue , he decided to divide the operation into half . he will write the problem and you are asked to solve it .. he wrote the problem and that is : { 8 }^{ x+1 }={ 9 }^{ x-2 } . Find x

If the Answer is Not An integer , Make it Approaches to the nearest integer.


The answer is 55.

Abdelrahman Othman Helal by Abdelrahman Othman Helal , 25, Egypt

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1 solution

Peter Adel
Feb 24, 2014

9 x − 2 = 8 x + 1 9^{x-2} = 8^{x+1} So

( x − 2 ) × L n ( 9 ) = ( x + 1 ) × L n ( 8 ) (x-2) \times Ln (9) = (x+1) \times Ln (8)

x L n ( 9 ) − 2 L n ( 9 ) = x L n ( 8 ) + L n ( 8 ) x Ln (9) - 2 Ln (9) = x Ln (8) + Ln (8)

and x = 2 L n ( 9 ) + L n ( 8 ) L n ( 9 ) − L n ( 8 ) x = \frac{2 Ln (9)+ Ln (8)}{Ln (9) - Ln (8)}

x = 54.964 x = 54.964

so after approximation

x = 55 x = \boxed{55}

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