Solve the Complex Logarithmic Equation: log₅x + log₅(x+2) + log₂5 - log₂2.5 = log₃7 × log₇9

To solve this problem, we will follow these steps:

• Step 1: Simplify the left-hand side using logarithm properties.
• Step 2: Simplify the right-hand side using change of base.
• Step 3: Equate simplified forms and solve for $x$.

Now, let's proceed:

Step 1: Simplify the left-hand side:
We can combine the logs as follows:
$\log_5 x + \log_5 (x+2) = \log_5 (x(x+2)) = \log_5 (x^2 + 2x).$
The constants are simplified as:
$\log_2 5 - \log_2 2.5 = \log_2 \left(\frac{5}{2.5}\right) = \log_2 2 = 1.$
Thus, the entire left-hand side becomes:
$\log_5 (x^2 + 2x) + 1.$

Step 2: Simplify the right-hand side:
$\log_3 7 \times \log_7 9$ can be written using the change of base formula:
$\log_3 7 = \frac{\log 7}{\log 3}$ and $\log_7 9 = \frac{\log 9}{\log 7}$. Multiplying these, we have:
$\frac{\log 9}{\log 3} = 2, \text{ since } \log 9 = \log 3^2 = 2 \log 3.$

Step 3: Equate and solve:
Equate the simplified versions:
$\log_5 (x^2 + 2x) + 1 = 2$
So, subtracting 1 from both sides:
$\log_5 (x^2 + 2x) = 1$
Taking antilogarithm, we find:
$x^2 + 2x = 5^1 = 5$

Rearrange to form a quadratic equation:
$x^2 + 2x - 5 = 0$

Step 4: Solve the quadratic equation:
Use the quadratic formula, where $a = 1$, $b = 2$, $c = -5$:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 20}}{2} = \frac{-2 \pm \sqrt{24}}{2} = \frac{-2 \pm 2\sqrt{6}}{2}$
$x = -1 \pm \sqrt{6}$

The valid answer must ensure $x + 2 > 0$, so $x = -1 + \sqrt{6}$.

Therefore, the solution to the problem is $x = -1 + \sqrt{6}$.