if log base a of x =c , and log base b of x = d, then what is the general statement that expresses log base (ab) of x in terms of c and d? plz help.
Use change of base formula:
$\displaystyle \log_a x=c\implies c=\frac{\ln x}{\ln a}\implies \ln a=\frac{\ln x}{c}$ and $\displaystyle \log_b x=d\implies d=\frac{\ln x}{\ln b}\implies \ln b=\frac{\ln x}{d}$.
Now, $\displaystyle \log_{ab}x=\frac{\ln x}{\ln\!\left(ab\right)}=\frac{\ln x}{\ln a+\ln b}=\frac{\ln x}{\frac{1}{c}\ln x+\frac{1}{d}\ln x}=\frac{1}{\frac{c+d}{cd}}=\frac{cd}{c+d}$
Does this make sense?
that was extremely helpful, but i did get lost at the part where
ln(x) was ÷ by the sum of (1/c ln(x) ) and (1/d ln(x) ). where or how did you get those expressions in the denominator?
Nevermind, now i get it!
thank youuu! >singing<