How many distinct solutions does the equation log10 x = 2 - log10 (x - 21) have?

Steps to Solve

1. Rewrite the logarithmic equation We start with the given equation: $$ \log_{10} x = 2 - \log_{10} (x - 21) $$

2. Combine the logarithms To simplify, we can rewrite the equation by moving the logarithmic term: $$ \log_{10} x + \log_{10} (x - 21) = 2 $$

Using the property of logarithms that states $\log_b a + \log_b c = \log_b(ac)$, we combine: $$ \log_{10} [x(x - 21)] = 2 $$

3. Exponentiate to remove the logarithm Next, we exponentiate both sides to eliminate the logarithm: $$ x(x - 21) = 10^2 $$ This simplifies to: $$ x^2 - 21x - 100 = 0 $$

4. Solve the quadratic equation We will use the quadratic formula, where $a = 1$, $b = -21$, and $c = -100$: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Calculating the discriminant: $$ b^2 - 4ac = (-21)^2 - 4(1)(-100) = 441 + 400 = 841 $$

The square root of 841 is 29.

5. Find the values of x Plugging the values back into the formula: $$ x = \frac{21 \pm 29}{2} $$

Calculating the two possible values:

1. ( x_1 = \frac{50}{2} = 25 )

2. ( x_2 = \frac{-8}{2} = -4 )

3. Check valid solutions The logarithm $\log_{10} (x - 21)$ requires ( x - 21 > 0 ), which implies ( x > 21 ). Thus, only take ( x_1 = 25 ) as a valid solution.