# Thread: Logarithms and patterns

1. ## Logarithms and patterns

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

2. loga (x) = c

logb (x) = d

logab (x) = ?

a^c = x and b^d = x

a = x ^ (1/c) and b = x ^ (1/d)

ab = x^ 1/c x ^ 1/d

= x ^ (d+c/cd)

logab (x) = d+c / cd

humm i might have flipped something

3. Hello, rubix!

Ya done good!

I'll put it in LaTex . . .

$\begin{array}{c}\log_a (x) \:=\: c \\ \log_b(x) \:=\:d\end{array}\qquad \log_{ab}(x) \:=\: ?$

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

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

Therefore: . $\log_{ab}(x) \:=\: \frac{c+d}{cd}$

Good work!

4. thnx for conforming Soroban...and thnx for nice well formatted answe.

cheers.

5. Thanks a lot for your help!