Question

if
${a}^{p} = {b}^{q} = {c}^{r}$
and
${b}^{2} = ac$
show that
$q = \frac{2rp}{r + p}$

plz answer fast

Answer 1

$\bold{Answer :}$

Let us take,

a^p = b^q = c^r = e^k (k ≠ 0)

Taking (log), we get

log (a^p) = log (b^q) = log (c^r) = log (e^k)

➩ p loga = q logb = r logc = k loge

➩ p loga = q logb = r logc = k

Then,

p loga = k ➩ loga = k/p

q logb = k ➩ logb = k/q

r logc = k ➩ logc = k/r

Given that,

b^2 = ac

Taking (log), we get

log (b^2) = log (ac)

➩ 2 logb = loga + logc

➩ 2 (k/q) = k/p + k/r

➩ 2/q = 1/p + 1/r

➩ 2/q = (r + p)/rp

➩ q/2 = rp/(r + p)

➩ q = 2rp/(r + p)

Therefore, q = 2rp/(r + p) [Proved]

Rules used :

log (ab) = loga + logb

log (a^b) = b loga

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