How to solve
Log (9x+5) - Log ((x^2)-1) = 1/2
16 16
(The 16 is from the logarithmic function : b) (y=log x)
b
Could anyone post the answer to this?
16 is supposed to be next to Log(16) but lower. This is for both the Logs in the equation.
To solve the equation:
Log (9x+5) - Log ((x^2)-1) = 1/2
16 16
We can start by using the logarithmic property known as the quotient rule, which states that:
Log (a) - Log (b) = Log (a/b)
Using this property, we can rewrite the equation as:
Log [(9x+5)/((x^2)-1)] = 1/2
16
Next, we can convert the equation into exponential form by raising both sides of the equation to the power of the base (16):
[(9x+5)/((x^2)-1)] = (16^(1/2))
Simplifying the right side, we have:
[(9x+5)/((x^2)-1)] = 4
Next, we can cross-multiply to get rid of the fraction:
(9x+5) = 4((x^2)-1)
Expanding the right side:
(9x+5) = 4x^2 - 4
Moving all terms to one side of the equation:
4x^2 - 9x - 4 - 5 = 0
Simplifying:
4x^2 - 9x - 9 = 0
At this point, we have a quadratic equation. To solve it, we can either factor it or use the quadratic formula.
Factoring:
The equation is not easily factorable, so let's use the quadratic formula to find the values of x.
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = -9, and c = -9. Plugging these values into the quadratic formula:
x = [(-(-9)) ± √((-9)^2 - 4(4)(-9))] / (2(4))
Simplifying:
x = (9 ± √(81 + 144)) / 8
-------------------
8
x = (9 ± √(225)) / 8
-------------
8
x = (9 ± 15) / 8
x1 = (9 + 15) / 8 = 24 / 8 = 3
x2 = (9 - 15) / 8 = -6 / 8 = -3/4
So the solutions to the equation are x = 3 and x = -3/4.