log_2(x-2)+log_2(x-3)=1

To solve the given equation, we need to combine the logarithmic terms using the properties of logarithms.

Remember that the sum of two logarithms with the same base is equivalent to the logarithm of the product of their arguments. So, we can rewrite the equation as:

log_2[(x-2)(x-3)] = 1

Now, we can convert the logarithmic form into exponential form:

2^1 = (x-2)(x-3)

Simplify the right side:

2 = x^2 - 5x + 6

Rearrange the equation to set it equal to zero:

x^2 - 5x + 4 = 0

Now, we need to solve this quadratic equation. We can either factor it or use the quadratic formula:

x = [5 ± sqrt((-5)^2 - 414)] / (2*1) x = [5 ± sqrt(25 - 16)] / 2 x = [5 ± sqrt(9)] / 2 x = [5 ± 3] / 2

Therefore, the solutions to the equation are:

x = 4 or x = 1

Check if these solutions are valid by plugging them back into the original equation.