Solve the logarithmic equation log x = 1 - log (x -3).
A) only -5
B) only 5
C) only -2
D) 5 and -2
E) -5 and -2

Question

Quizplus · Accepted Answer

To solve the equation log x = 1 - log (x - 3), we can first combine the logarithms on one side. This gives us log x + log (x - 3) = 1. Using the property of logarithms that log a + log b = log ab, we can combine the left side to get log [x(x - 3)] = 1. This simplifies to log (x^2 - 3x) = 1. To remove the logarithm, we use the fact that if log a = b, then 10^b = a. So, 10^1 = x^2 - 3x, which simplifies to x^2 - 3x - 10 = 0. Factoring this quadratic equation, we get (x - 5)(x + 2) = 0, which gives us x = 5 and x = -2. However, if we substitute x = -2 back into the original equation, it doesn't work because log (-2) and log (-2 - 3) are undefined. Therefore, the only solution that works in the original equation is x = 5.

Solve the Logarithmic Equation Log X = 1 - Log