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.

So we are looking at

log16 (9x+5) - log16</sub (x^2 - 1) = 1/2
log16</sub [(9x+5)/(x^2-1)] = 1/2
(9x+5)/(x^2-1) = 16^(1/2)
(9x+5)/(x^2-1) = 4
4x^2 - 4 = 9x + 5
4x^2 - 9x - 9 = 0
(x-3)(4x + 3) = 0
x = 3 or x = -3/4

but when x = -3/4, the first log term is undefined (we can't take a log of a negative)

so x = 3

To solve the equation, we will use the properties of logarithms. First, let's rewrite the equation with the given information:

log₆ (9x + 5) / log₆ ((x² - 1) = 1/2

Now, we can use the quotient rule of logarithms, which states that logₐ (b/c) = logₐ (b) - logₐ (c). Applying this rule, the equation becomes:

log₆ (9x + 5) - log₆ (x² - 1) = 1/2

Next, we can use the power rule of logarithms, which states that logₐ (b^c) = c * logₐ (b). Applying this rule to the left side of the equation, we have:

log₆ ((9x + 5) / (x² - 1)) = 1/2

To simplify further, we will convert the equation from logarithmic form to exponential form. In exponential form, a logarithm with base b is equivalent to b raised to the power of the logarithm's result. Applying this to our equation, we have:

(9x + 5) / (x² - 1) = 6^(1/2)

To solve for x, we'll start by squaring both sides of the equation, resulting in:

(9x + 5) / (x² - 1) = 6

Next, we'll cross-multiply to eliminate the fraction:

(9x + 5) = 6(x² - 1)

Expanding the right side of the equation:

9x + 5 = 6x² - 6

Bringing all terms to one side of the equation:

6x² - 9x - 11 = 0

To solve this quadratic equation, we can use factoring, the quadratic formula, or completing the square methods. Factoring may not be feasible for this equation, so let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

For our equation, a = 6, b = -9, and c = -11. Substituting these values into the quadratic formula, we get:

x = (-(-9) ± √((-9)² - 4(6)(-11))) / (2(6))
x = (9 ± √(81 + 264)) / 12
x = (9 ± √345) / 12

So, the two solutions to the equation are:
x₁ = (9 + √345) / 12
x₂ = (9 - √345) / 12