To solve this equation, we need to use the properties of logarithms.
First, we can combine the two logarithmic terms on the left side of the equation using the quotient rule of logarithms. This rule states that log_a(b) - log_a(c) = log_a(b/c).
So, we have:
log_7((2x^2 - 7x + 6)/(x - 2)) = log_7(x)
Now, we can remove the logarithms by exponentiating both sides with base 7.
This gives us:
(2x^2 - 7x + 6)/(x - 2) = x
Now, we can simplify this equation by multiplying both sides by (x - 2) to get rid of the fraction:
2x^2 - 7x + 6 = x(x - 2) 2x^2 - 7x + 6 = x^2 - 2x
Next, we can combine like terms and set the equation equal to zero:
x^2 - 5x + 6 = 0
Now, we can factor this quadratic equation:
(x - 2)(x - 3) = 0
Setting each factor equal to zero gives us the solutions:
x - 2 = 0 or x - 3 = 0
Therefore, the solutions to the original equation are x = 2 or x = 3.
To solve this equation, we need to use the properties of logarithms.
First, we can combine the two logarithmic terms on the left side of the equation using the quotient rule of logarithms. This rule states that log_a(b) - log_a(c) = log_a(b/c).
So, we have:
log_7((2x^2 - 7x + 6)/(x - 2)) = log_7(x)
Now, we can remove the logarithms by exponentiating both sides with base 7.
This gives us:
(2x^2 - 7x + 6)/(x - 2) = x
Now, we can simplify this equation by multiplying both sides by (x - 2) to get rid of the fraction:
2x^2 - 7x + 6 = x(x - 2)
2x^2 - 7x + 6 = x^2 - 2x
Next, we can combine like terms and set the equation equal to zero:
x^2 - 5x + 6 = 0
Now, we can factor this quadratic equation:
(x - 2)(x - 3) = 0
Setting each factor equal to zero gives us the solutions:
x - 2 = 0 or x - 3 = 0
Therefore, the solutions to the original equation are x = 2 or x = 3.