Question

help me please!
if f(z)= log z, show that f(xy)=f(x)+f(y).
kindly answer this. i really neeed this right now! thank you a lot :)

Answer 1

\(f(xy)=\log(xy)\)
\(f(x) = \log x\)
\(f(y) =\log y\)
So you want to show \(\log(xy)=\log x+\log y\)

say \(p = \log x\) and \(q=\log y\)
then \( 10^p=x\) and \(10^q=y\)
So,
\(xy=
10^p\times 10^q=10^{p+q}\)

log both sides:
\(\log(xy)=\log(10^{p+q})\)
\(\log(xy)=(p+q)\log 10\)
\(\log(xy)=p+q\), since \(\log 10=1\)
\(\log(xy)=\log x +\log y\), since this is how we defined \(p\) and \(q\) above