To find common difference of a logarithmic AP.

Question

Ques: If $a,b,c$ are in GP and $\log_ba,\log_cb,\log_ac$ are in AP. Then find the common difference of AP.

Here's what I did: $\Rightarrow b^2=ac$
$\Rightarrow 2\log b=\log a+\log c$ i.e. $\log a,\log b,\log c$ are in AP.

Now using the AP we get, $2\log_cb=\log_ba+\log_ac$ and we have to find
$\log_ac-\log_cb$ or $\log_cb-\log_ba$.

I have problem coping with the bases. Help me I know the answer(it is $3/2$). Maybe I'm a bit weak with logarithms, thats why I can't solve this.
Please Help me. Thank you.

Answer 1

Hint:

WLOG $b=ar,c=ar^2$

$$P=\log_ba=\dfrac{\log a}{\log a+\log r}$$

$$Q=\log_cb=\dfrac{\log a+\log r}{\log a+2\log r}$$

$$R=\log_ac=\dfrac{\log a+2\log r}{\log a}$$

Use $P+R=2Q$ to find the relation between $\log a,\log r$