Log 3(x-3)+log3 x=log3 4

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To solve this logarithmic equation, we need to combine the two logarithms on the left side of the equation using the product rule of logarithms.

Log a + Log b = Log (a * b)

So, we have:

Log3(x-3) + Log3 x = Log3 4

Applying the product rule:

Log3[(x-3)*x] = Log3 4

Now, simplify the expression inside the logarithm:

Log3(x^2 - 3x) = Log3 4

Since the base of the logarithm on both sides is the same (base 3), we can drop the logarithm and equate the expressions inside the logarithm:

x^2 - 3x = 4

Rearrange the equation into a quadratic form:

x^2 - 3x - 4 = 0

Now, we can factorize the quadratic equation:

(x - 4)(x + 1) = 0

Setting each factor to zero:

x - 4 = 0 or x + 1 = 0

Solving for x:

x = 4 or x = -1

However, we cannot take the logarithm of a negative number, so the solution for x is:

x = 4