Question

determine whether the following statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement.
if \\( \log (x + 3) = 6 \\), then \\( e^6 = x + 3 \\).

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\( \bigcirc \\) a. the statement is true.
\\( \bigcirc \\) b. the statement is false. the correct statement is if \\( \log (x + 3) = 6 \\), then \\( \square = x + 3 \\).
(do not simplify.)

Explanation:

Step1: Recall Logarithm Definition

The notation \(\log(x + 3)\) without a base specified usually implies the common logarithm (base 10), not the natural logarithm (base \(e\)). The definition of a logarithm \(\log_b(y)=k\) means \(b^k = y\). For common logarithm, \(b = 10\), so if \(\log(x + 3)=6\) (base 10), then \(10^6=x + 3\), not \(e^6\).

Step2: Correct the Statement

Since the original statement used \(\log\) (implied base 10) and set \(e^6=x + 3\), it's false. The correct base - raised - to - the - power should be with base 10, so \(10^6=x + 3\).

Answer:

B. The statement is false. The correct statement is if \(\log(x + 3)=6\), then \(10^{6}=x + 3\).