Simplifying Logarithmic Expressions to \log_x{y}

[thereddevils]

Homework Statement

Simplify $$(\log_a{b})(\log_b{c})(\log_c{d})$$ in terms of $$\log_x{y}$$

Homework Equations

The Attempt at a Solution

well , the furthest i can reach is

$$\frac{\log_xd}{\log_xa}$$ , how do i express in terms of $$\log_x{y}$$

 

[HallsofIvy]

Since there are no "x" or "y" in the problem I think you are misunderstanding.

"in term so $log_x y$" simply means "in terms of a single logarithm"

You could, for example, put everything in terms of a logarithm base a:
$$log_b c= \frac{log_a c}{log_a b}$$
and
$$log_c d= \frac{log_a d}{log_a c}$$
so
$$(log_a b)(log_b c)(log_c d)= (log_a b)\frac{log_a c}{log_a b}\frac{log_a d}{log_a c}= log_a d$$

 

[thereddevils]

Since there are no "x" or "y" in the problem I think you are misunderstanding.

"in term so $log_x y$" simply means "in terms of a single logarithm"

You could, for example, put everything in terms of a logarithm base a:
$$log_b c= \frac{log_a c}{log_a b}$$
and
$$log_c d= \frac{log_a d}{log_a c}$$
so
$$(log_a b)(log_b c)(log_c d)= (log_a b)\frac{log_a c}{log_a b}\frac{log_a d}{log_a c}= log_a d$$

thanks !