what will be value of x in terms of logrithims
Answers
Question :-
log 2, log (2^x - 1) , log (2^x + 3) be three consecutive terms of AP then find the value of x.
Answer :-
log₂5
Solution :-
log 2, log (2^x - 1) , log (2^x + 3) are in AP
Here,
- a = log 2
- a2 = log (2^x - 1)
- a3 = log (2^x + 3)
Since there are in AP,
==> Common difference = a2 - a = a3 - a2
==> log (2^x - 1) - log 2 = log (2^x + 3) - log (2^x - 1)
==> log (2^x - 1) + log (2^x - 1) = log 2 + log (2^x + 3)
Using product rule log a + log b = log ab
==> log (2^x - 1)² = log 2(2^x + 3)
Comparing on both sides
==> (2^x - 1)² = 2(2^x + 3)
Substituting 2^x = y in the above equation
==> (y - 1)² = 2(y + 3)
==> y² - 2y + 1 = 2y + 6
==> y² - 2y - 2y + 1 - 6 = 0
==> y² - 4y - 5 = 0
==> y² - 5y + y - 5 = 0
==> y(y - 5) + 1(y - 5) = 0
==> (y + 1)(y - 5) = 0
==> y + 1 = 0 or y - 5 = 0
==> y = - 1 or y = 5
y = - 1 is not possible
==> y = 5
==> 2^x = 5
Writing it in exponential form
==> log₂5 = x
[ Because if a^x = N then logₐN = x ]
==> x = log₂5
Therefore the value of x is log₂5.
Answer:
log₂ 5
Step-by-step explanation:
Given :
log 2 , log ( 2ˣ -1 ) and log ( 2ˣ + 3 ) are in A.P.
We know if three numbers a , b and c are in A.P. then
2 b = a + c
= > 2 log ( 2ˣ -1 ) = log ( 2ˣ + 3 ) + log ( 2 )
Using properties of logorithm :
= > log ( 2ˣ -1 )² = log [ ( 2ˣ + 3 ) × ( 2 ) ]
Comparing both side we get :
= > ( 2ˣ -1 )² = 2 . ( 2ˣ + 3 )
Let 2ˣ = k
= > k² + 1 - 2 k = 2 k + 6
= > k² - 4 k - 5 = 0
= > k² - 5 k + k - 5 = 0
= > ( k - 5 ) ( k + 1 ) = 0
= > k = 5 OR k = - 1
Since log is applicable for positive number
Therefore - 1 is incorrect. Using other value
2ˣ = k
= > 2ˣ = 5
= > log₂ 5 = x