To determine the correct sequence of steps for deriving the difference quotient for the function [tex]\( f(x) = 3 - \log x \)[/tex], let's follow the step-by-step method.
The difference quotient formula is given by:
[tex]\[ \frac{f(x+h) - f(x)}{h} \][/tex]
Substituting [tex]\( f(x) = 3 - \log x \)[/tex] and [tex]\( f(x+h) = 3 - \log (x+h) \)[/tex] into the difference quotient, we get:
[tex]\[
\frac{(3 - \log (x+h)) - (3 - \log x)}{h}
\][/tex]
Next, simplify the expression inside the numerator:
[tex]\[
\frac{3 - \log (x+h) - 3 + \log x}{h}
\][/tex]
Combine like terms (the constant 3's cancel each other out):
[tex]\[
\frac{-\log (x+h) + \log x}{h}
\][/tex]
Reordering the terms in the numerator gives:
[tex]\[
\frac{\log x - \log (x+h)}{h}
\][/tex]
Thus, the correct sequence of steps is as follows:
[tex]\[
\frac{f(x+h)-f(x)}{h}
\][/tex]
[tex]\[
= \frac{(3-\log (x+h))-(3-\log x)}{h}
\][/tex]
[tex]\[
= \frac{3-\log (x+h)-3+\log x}{h}
\][/tex]
[tex]\[
= \frac{-\log (x+h)+\log x}{h} ; \quad h \neq 0
\][/tex]
Therefore, the first sequence of steps is the correct one.
