If a, b, c are in G.P, then loga 10, logb 10 and logc 10 are in(a) A.P

1.	If a, b, c are in G.P, then loga 10, logb 10 and logc 10 are in(a) A.P (b) G.P (c) H.P. (d) None of these
Answer» (c) H.P. a, b, c are in G.P ⇒ b² = ac ⇒ \(\frac{1}{\text{log}_a\,10}\) + \(\frac{1}{\text{log}_c\,10}\) = log₁₀ a + log₁₀ c = log₁₀ (ac) = = log₁₀ b² = 2 log₁₀ b = \(\frac{2}{\text{log}_b\,10}\) ⇒ \(\frac{1}{\text{log}_a\,10}\), \(\frac{1}{\text{log}_b\,10}\), \(\frac{1}{\text{log}_c\,10}\) are in A.P. ⇒ log_a 10, log_b 10, log_c 10 are in H.P.

If a, b, c are in G.P, then loga 10, logb 10 and logc 10 are in(a) A.P (b) G.P (c) H.P. (d) None of these

Answer»

(c) H.P.

a, b, c are in G.P ⇒ b² = ac

⇒ \(\frac{1}{\text{log}_a\,10}\) + \(\frac{1}{\text{log}_c\,10}\) = log₁₀ a + log₁₀ c = log₁₀ (ac)

= = log₁₀ b² = 2 log₁₀ b = \(\frac{2}{\text{log}_b\,10}\)

⇒ \(\frac{1}{\text{log}_a\,10}\), \(\frac{1}{\text{log}_b\,10}\), \(\frac{1}{\text{log}_c\,10}\) are in A.P.

⇒ log_a 10, log_b 10, log_c 10 are in H.P.