Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer. log x + log (x + 2) = log 3 solve the equation to find the solution set. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is {}. (simplify your answer. use a comma to separate answers as needed.) b. there are infinitely many solutions. c. there is no solution

Explanation:

Step1: Apply log product rule

Using the property \(\log a+\log b = \log(ab)\), we rewrite the left - hand side of the equation \(\log x+\log(x + 2)=\log3\) as \(\log[x(x + 2)]=\log3\).

Step2: Use log equality property

If \(\log m=\log n\) (where the logarithms have the same base), then \(m = n\) (for \(m>0\) and \(n>0\)). So we have the equation \(x(x + 2)=3\).

Step3: Expand and solve quadratic

Expand the left - hand side: \(x^{2}+2x=3\). Rearrange it to the standard quadratic form \(x^{2}+2x - 3=0\). Factor the quadratic: \((x + 3)(x - 1)=0\).
Set each factor equal to zero: \(x+3 = 0\) gives \(x=-3\) and \(x - 1=0\) gives \(x = 1\).

Step4: Check domain

For the original logarithmic expressions \(\log x\) and \(\log(x + 2)\), the argument of a logarithm must be positive.

• For \(x=-3\): \(\log(-3)\) and \(\log(-3 + 2)=\log(-1)\) are undefined (since the argument of a logarithm cannot be negative).
• For \(x = 1\): \(\log(1)=0\) and \(\log(1 + 2)=\log(3)\) are defined (since \(1>0\) and \(1 + 2=3>0\)).

Answer:

A. The solution set is \(\{1\}\)