log base 2 x + log base 2 (x-7)=3 solving steps?

Hmm. I read it as

x(x-7) = 8

Oddly enough, the answer is the same!

To solve the equation log base 2 x + log base 2 (x-7) = 3, we can use the properties of logarithms. The sum of logarithms with the same base can be obtained by multiplying the numbers inside the logarithms. Therefore, we can rewrite the equation as:

log base 2 (x) * (x-7) = 3

Next, we can use the exponentiation property of logarithms, where if log base b (a) = c, then b^c = a. Applying this property, we can rewrite the equation as:

2^3 = x * (x-7)

Now, simplify the equation:

8 = x^2 - 7x

Rearrange the equation to set it equal to zero:

x^2 - 7x - 8 = 0

Now, we can factorize the quadratic equation:

(x - 8)(x + 1) = 0

Setting each factor equal to zero, we have:

x - 8 = 0 or x + 1 = 0

Solving for x in each equation gives us two solutions:

x = 8 or x = -1

Therefore, the solutions to the equation log base 2 x + log base 2 (x-7) = 3 are x = 8 and x = -1.

To solve the equation log base 2 x + log base 2 (x-7) = 3, you can use the properties of logarithms.

Step 1: Combine the logarithms using the product rule of logarithms. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
log base 2 x + log base 2 (x-7) = log base 2 (x(x-7))

Step 2: Apply the power rule of logarithms to the right-hand side. The power rule states that the logarithm of a number raised to a power is equal to the product of the power and the logarithm of the number. In this case, we have:
log base 2 (x(x-7)) = log base 2 (2^3)
log base 2 (x(x-7)) = 3

Step 3: Rewrite the equation in exponential form. In exponential form, the logarithmic equation becomes:
2^3 = x(x-7)

Step 4: Simplify the right-hand side of the equation by expanding the equation:
8 = x^2 - 7x

Step 5: Move all terms to one side of the equation to obtain a quadratic equation:
x^2 - 7x - 8 = 0

Step 6: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula:
This equation can be factored as (x - 8)(x + 1) = 0
Setting each factor equal to zero, we get two solutions:
x - 8 = 0 => x = 8
x + 1 = 0 => x = -1

So, the possible values of x that satisfy the equation are x = 8 and x = -1.

log2 x + log2 (x-7) = 3

log2( x/(x-7) ) = 3

x/(x-7) = 2^3 = 8
8x - 56 = x
7x = 56
x = 8