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 63\\) 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. \\(\bigcirc\\) a the solution set is \\(\boxed{}\\) (simplify your answer. use a comma to separate answers as needed) \\(\bigcirc\\) b there are infinitely many solutions \\(\bigcirc\\) c there is no solution

Explanation:

Step1: Apply Logarithm Product Rule

The product rule of logarithms states that \(\log a + \log b = \log(ab)\). So, for the equation \(\log x+\log(x + 2)=\log63\), we can combine the left - hand side:
\(\log[x(x + 2)]=\log63\)

Step2: Use Logarithm One - to - One Property

If \(\log_aM=\log_aN\) (where \(a>0,a
eq1,M>0,N>0\)), then \(M = N\). Since the base of the logarithm (assuming base 10, but the property holds for any valid base) is the same on both sides, we have:
\(x(x + 2)=63\)
Expand the left - hand side: \(x^{2}+2x=63\)
Rearrange the equation to get a quadratic equation: \(x^{2}+2x - 63 = 0\)

Step3: Solve the Quadratic Equation

We can factor the quadratic equation \(x^{2}+2x - 63\). We need two numbers that multiply to \(- 63\) and add up to \(2\). The numbers are \(9\) and \(-7\). So,
\(x^{2}+2x - 63=(x + 9)(x - 7)=0\)
Set each factor equal to zero:
\(x+9 = 0\) gives \(x=-9\)
\(x - 7=0\) gives \(x = 7\)

Step4: Check the Domain of Logarithms

For a logarithm \(\log y\), \(y>0\).

• For \(x=-9\):
• When \(x=-9\), \(\log x=\log(-9)\) is undefined (since the argument of a logarithm must be positive).
• When \(x=-9\), \(x + 2=-9 + 2=-7\), and \(\log(x + 2)=\log(-7)\) is also undefined.
• For \(x = 7\):
• When \(x = 7\), \(\log x=\log7\) (defined, since \(7>0\)) and \(x + 2=7 + 2 = 9\), \(\log(x + 2)=\log9\) (defined, since \(9>0\)). And \(\log7+\log9=\log(7\times9)=\log63\), which satisfies the original equation.

Answer:

\(7\)