What is the exact solution of the equation log(x - 21) + log(x) = 2?

To solve the equation log(x - 21) + log(x) = 2, we will use the properties of logarithms to combine the logs. The property that states log(a) + log(b) = log(a * b) allows us to rewrite the equation as:

log((x - 21) * x) = 2.

Next, to eliminate the logarithm, we can exponentiate both sides. This means that we rewrite the equation in its exponential form, which gives us:

(x - 21) * x = 10^2,

or

(x - 21) * x = 100.

Expanding the left side results in:

x^2 - 21x = 100.

To set this quadratic equation to zero, we can rearrange it:

x^2 - 21x - 100 = 0.

Next, we can factor this quadratic equation. We need two numbers that multiply to -100 and add to -21. These numbers are -25 and 4. Thus, we can factor the equation as:

(x - 25)(x + 4) = 0.

Setting each factor to zero gives us the potential solutions:

x - 25 = 0