To solve this logarithmic equation, we can use the properties of logarithms to condense the left side of the equation into a single logarithmic term:
log3 x + log3(x-2) = log3(2x-3)
Using the properties of logarithms, we can combine the two logarithms on the left side by multiplying them:
log3 (x(x-2)) = log3(2x-3)
Now we have a single logarithmic term on both sides of the equation. Since the bases are the same (base 3), we can remove the logarithms and set the expressions inside the logarithms equal to each other:
x(x-2) = 2x-3
Now, we can expand the left side of the equation:
x^2 - 2x = 2x - 3
Next, we can rearrange the terms to set the equation equal to zero:
x^2 - 4x + 3 = 0
Now, we can factor the quadratic equation:
(x-3)(x-1) = 0
Setting each factor equal to zero gives us the possible solutions:
x-3 = 0 --> x = 3 x-1 = 0 --> x = 1
Therefore, the solutions to the equation log3 x + log3(x-2) = log3(2x-3) are x = 1 and x = 3.
To solve this logarithmic equation, we can use the properties of logarithms to condense the left side of the equation into a single logarithmic term:
log3 x + log3(x-2) = log3(2x-3)
Using the properties of logarithms, we can combine the two logarithms on the left side by multiplying them:
log3 (x(x-2)) = log3(2x-3)
Now we have a single logarithmic term on both sides of the equation. Since the bases are the same (base 3), we can remove the logarithms and set the expressions inside the logarithms equal to each other:
x(x-2) = 2x-3
Now, we can expand the left side of the equation:
x^2 - 2x = 2x - 3
Next, we can rearrange the terms to set the equation equal to zero:
x^2 - 4x + 3 = 0
Now, we can factor the quadratic equation:
(x-3)(x-1) = 0
Setting each factor equal to zero gives us the possible solutions:
x-3 = 0 --> x = 3
x-1 = 0 --> x = 1
Therefore, the solutions to the equation log3 x + log3(x-2) = log3(2x-3) are x = 1 and x = 3.