To solve the given problem, we need to use the change of base formula for logarithms. The change of base formula states that for any positive numbers [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] (with [tex]\( a \neq 1 \)[/tex] and [tex]\( b \neq 1 \)[/tex]), the logarithm of [tex]\( a \)[/tex] to the base [tex]\( b \)[/tex] can be expressed as:
[tex]\[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \][/tex]
Here, we want to apply this change of base formula to the logarithm [tex]\(\log_4(x+2)\)[/tex]. According to the change of base formula, we can express [tex]\(\log_4(x+2)\)[/tex] using a common logarithm (base 10) or a natural logarithm (base [tex]\( e \)[/tex]). Let's use the common logarithm (base 10) for simplicity:
[tex]\[ \log_4(x+2) = \frac{\log(x+2)}{\log(4)} \][/tex]
This shows that [tex]\(\log_4(x+2)\)[/tex] can be rewritten as the ratio of the logarithm (base 10) of [tex]\( x + 2 \)[/tex] to the logarithm (base 10) of 4.
Given the options:
1. [tex]\(\frac{\log (x+2)}{\log 4}\)[/tex]
2. [tex]\(\frac{\log 4}{\log (x+2)}\)[/tex]
3. [tex]\(\frac{\log 4}{\log x+2}\)[/tex]
4. [tex]\(\frac{\log x+2}{\log 4}\)[/tex]
The correct expression that results when the change of base formula is applied to [tex]\(\log_4(x+2)\)[/tex] is:
[tex]\[ \boxed{\frac{\log (x+2)}{\log 4}} \][/tex]
Therefore, the correct answer is option 1.
