General Statement

Bzaher · #1 $Old$ November 17th, 2009, 05:08 PM

Let Logax = c and Logbx = d

I need to find the general statement that expresses Logabx in the form of c and d

Soroban · #2 $Old$ November 17th, 2009, 06:36 PM

Hello, Bzaher!

Quote:

Let: . $\begin{array}{ccc}\log_ax \:=\:c \\ \log_bx \:=\:d \end{array}$

Find the $\log_{ab}x$ in terms of $c$ and $d$ .

$\begin{array}{cccccc}\log_ax \:=\:c & \Rightarrow& a^c \:=\:x & \Rightarrow & a \:=\:x^{\frac{1}{c}} & [1] \\ \log_bx\:=\:d & \Rightarrow & b^d \:=\:x & \Rightarrow & b \:=\:x^{\frac{1}{d}} & [2] \end{array}$

Multiply [1] and [2]: . $ab \:=\:x^{\frac{1}{c}}\cdot x^{\frac{1}{d}} \;=\;x^{\frac{1}{c} + \frac{1}{d}} \quad\Rightarrow\quad ab \;=\;x^{\frac{c+d}{cd}}$

Therefore: . $ab^{\frac{cd}{c+d}} \;=\;x \quad\Rightarrow\quad \log_{ab}x \;=\;\frac{cd}{c+d}$

Bzaher · #3 $Old$ November 17th, 2009, 09:23 PM

Thank you so much for the answer, but why is it x to the power of 1/c and 1/d