solve the equation

loga(x-5)- loga (x+9)=loga(x-7)-loga(x+9)

all a's are lower

please show work

collect logs

log((x-5)/(x+9)) = log((x-7)/(x+9))
(x-5)/(x+9) = (x-7)/(x-9)
assuming x ≠ -9,
x-5 = x-7
5 = 7

no solution - typo?
even easier, add log(x+9) to both sides to get

log(x-5) = log(x-7)
x-5 = x-7
no solution

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To solve the equation loga(x-5) - loga(x+9) = loga(x-7) - loga(x+9), you can follow these steps:

Step 1: Use the logarithmic rules that state:
- loga(b) - loga(c) = loga(b/c)
- loga(b) = loga(c) if and only if b = c.

Applying these rules, we can rewrite the equation as:
loga((x-5)/(x+9)) = loga((x-7)/(x+9))

Step 2: Since the bases of the logarithms are the same (a), we can equate the two arguments inside the logarithms:
(x-5)/(x+9) = (x-7)/(x+9)

Step 3: Now we can solve the equation by cross-multiplying:
(x-5)(x+9) = (x-7)(x+9)

Expanding both sides gives:
x^2 + 4x - 45 = x^2 + 2x - 63

Step 4: Simplify the equation:
x^2 + 4x - 45 - x^2 - 2x + 63 = 0
2x + 18 = 0

Step 5: Solve for x by isolating the variable:
2x = -18
x = -9

Thus, the value of x that satisfies the given equation is x = -9.