Here is a quetion I couldn't solve.
Let loga (x) = c and logb (x) = d.
Find the general statement that expresses logab (x), in terms of c and d.
I would appreciate your help, thanks a lot,
Amine
Hello, rubix!
Ya done good!
I'll put it in LaTex . . .
$\displaystyle \begin{array}{c}\log_a (x) \:=\: c \\ \log_b(x) \:=\:d\end{array}\qquad \log_{ab}(x) \:=\: ?$
$\displaystyle \begin{array}{ccccc}a^c \:= \:x & \Rightarrow & a \:=\:x^{\frac{1}{x}} & [1] \\ b^d \:=\: x & \Rightarrow & b \:=\:x^{\frac{1}{d}} & [2]\end{array}$
Multiply [1] and [2]: .$\displaystyle ab \:=\:x^{\frac{1}{c}}\cdot x^{\frac{1}{d}} \;=\;x^{\frac{d+c}{cd}}$
Therefore: .$\displaystyle \log_{ab}(x) \:=\: \frac{c+d}{cd}$
Good work!