If y2 = xz and px = qy = rz, then what is the value of log p . log r?

Concept

When several exponential expressions are all set equal to the same value (here px = qy = rz, with p, q, r positive reals other than 1 so every log below is defined and nonzero), taking the logarithm of each side converts every exponent into a reciprocal expression involving that common log value. By the power rule log(mn) = n . log m, this turns any relationship among the exponents (such as y2 = xz) into an equivalent relationship purely among log p, log q, and log r.

Application

1. Let px = qy = rz = k. Taking the log of each part gives log px = log qy = log rz = log k.

2. By the power rule, this becomes x . log p = y . log q = z . log r = log k.

3. Solving each equation for its exponent: x = log k / log p, y = log k / log q, and z = log k / log r.

4. Substitute these into the given relation y2 = xz: (log k / log q)2 = (log k / log p) × (log k / log r).

5. Both sides carry a (log k)2 factor in the numerator, so it cancels: 1 / (log q)2 = 1 / (log p . log r).

6. Taking the reciprocal of both sides gives log p . log r = (log q)2.

Cross-check

Pick x = 1, y = 2, z = 4, which satisfies y2 = xz since 22 = 1 × 4 = 4. Let p1 = q2 = r4 = 16, so p = 16, q = 4, and r = 2. Using base-10 logs: log p = log 16 ~ 1.204, log q = log 4 ~ 0.602, log r = log 2 ~ 0.301. Then (log q)2 ~ 0.6022 ~ 0.362, and log p . log r ~ 1.204 × 0.301 ~ 0.362 -- the two sides match, confirming log p . log r = (log q)2.