Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact answer. \\(\log_{3}(x + 3) = -2\\)
rewrite the given equation without logarithms. do not solve for x

Explanation:

Step1: Recall logarithm definition

The logarithmic equation \(\log_b a = c\) can be rewritten in exponential form as \(b^c = a\), where \(b>0\), \(b
eq1\), and \(a>0\).

Step2: Apply the definition

For the equation \(\log_3(x + 3)=-2\), here \(b = 3\), \(c=-2\), and \(a=x + 3\). Using the exponential form, we get \(3^{-2}=x + 3\).

Answer:

\(3^{-2}=x + 3\) (or equivalently \(\frac{1}{9}=x + 3\) if simplified, but the form without simplifying the exponent is also correct as per the "rewrite without logarithms" request)