Solve this equation. Verify your solution using a graphing utility.
log_a(x - 1) - log_a(x + 6) = log_a(x - 2) - log_a(x + 3)
To solve the equation log_a(x - 1) - log_a(x + 6) = log_a(x - 2) - log_a(x + 3), we can simplify and rearrange the terms using logarithmic rules.
First, we can combine the logarithms on both sides of the equation using the quotient rule of logarithms, which states that log_a(b) - log_a(c) = log_a(b/c). Applying this rule, we have:
log_a((x - 1)/(x + 6)) = log_a((x - 2)/(x + 3))
Next, we can eliminate the logarithms by equating the arguments of the logarithms, since log_a(b) = log_a(c) if and only if b = c. Therefore, we have:
(x - 1)/(x + 6) = (x - 2)/(x + 3)
Now, we can solve this equation for x. We can start by cross-multiplying:
(x - 1)(x + 3) = (x - 2)(x + 6)
Expanding both sides gives:
x^2 + 3x - x - 3 = x^2 + 6x - 2x - 12
Simplifying further:
x^2 + 2x - 3 = x^2 + 4x - 12
Subtracting x^2 from both sides:
2x - 3 = 4x - 12
Moving the variables to one side:
2x - 4x = -12 + 3
-2x = -9
Dividing by -2:
x = 9/2 or x = 4.5
Therefore, the solution to the equation is x = 9/2 or x = 4.5.
To verify the solution using a graphing utility, we can plot the graphs of both sides of the equation and check where they intersect. If the graphs intersect at x = 9/2 or x = 4.5, then the solution is correct.
You can use online graphing calculators or software like Desmos, GeoGebra, or Wolfram Alpha to graph the equation and verify the solution visually.