Logarithmic Expression are solved using Logarithmic rules/identities and find unknown value of a variable x.
Topic : Solving for x using Logarithm
Below problem explains use of logarithmic identities/properties and also quadratic equation.
Problem : log 3 (x + 11) + log 3(x + 5) = 3, Solve for x
Solution :
We have the property of log as
log a + log b = log (ab)
So, log 3 (x + 11) + log 3(x + 5) = log (3(x + 11)3(x + 5)) = log 9(x² + 16x + 55)
Thus we have ; log 9(x² + 16x + 55) = 3
as we have the base of given logarithmic expression 10,
So 9(x² + 16x + 55) = 10³ (by the definition of logarithm identity)
9x² + 144x + 495 = 1000
9x² + 144x + 495 - 1000 = 0
9x² + 144x - 505 = 0
x = -144 ±√[144² - 4 (9)(-505)] / 2(9)
= -144 ±√[20736 + 18180] / 18
= -144 ±√[38916] / 18
= -144 ± 197.27 / 18
Thus either x = 2.96 or -18.96
But as log of a negative number is not defined, we discard x = -18.96
hence the answer is x = 2.96
If you have any queries please write to us.
Topic : Solving for x using Logarithm
Below problem explains use of logarithmic identities/properties and also quadratic equation.
Problem : log 3 (x + 11) + log 3(x + 5) = 3, Solve for x
Solution :
We have the property of log as
log a + log b = log (ab)
So, log 3 (x + 11) + log 3(x + 5) = log (3(x + 11)3(x + 5)) = log 9(x² + 16x + 55)
Thus we have ; log 9(x² + 16x + 55) = 3
as we have the base of given logarithmic expression 10,
So 9(x² + 16x + 55) = 10³ (by the definition of logarithm identity)
9x² + 144x + 495 = 1000
9x² + 144x + 495 - 1000 = 0
9x² + 144x - 505 = 0
x = -144 ±√[144² - 4 (9)(-505)] / 2(9)
= -144 ±√[20736 + 18180] / 18
= -144 ±√[38916] / 18
= -144 ± 197.27 / 18
Thus either x = 2.96 or -18.96
But as log of a negative number is not defined, we discard x = -18.96
hence the answer is x = 2.96
If you have any queries please write to us.
