An algebra problem by Razif FA

Algebra Level 3

$\log_a b , \log_b c , \log_c d$

Given that the three positive integers above formed a geometric progression . If $a=2$ and $d=128$ , find the second term of this progression.

2

8

\sqrt[3]{6}

4

\sqrt7

\sqrt[3]{7}

\sqrt6

32

1 solution

Hung Woei Neoh
Jun 13, 2016

Given that $\log_a b,\log_b c,\log_c d$ form a GP

$r=\dfrac{\log_b c}{\log_a b} = \dfrac{\log_c d}{\log_b c}\\ (\log_b c)^2 = (\log_a b)(\log_c d)\\ (\log_b c)^2 = (\log_a b)\left(\dfrac{\log_a d}{\log_a c}\right)\\ (\log_b c)^2 \left(\dfrac{\log_a c}{\log_a b}\right)= \log_a d\\ (\log_b c)^2(\log_b c) = \log_2 128\\ (\log_b c)^3 = \log_2 2^7\\ (\log_b c)^3 =7\\ T_2 = \log_b c = \boxed{\sqrt[3]{7}}$

An algebra problem by Razif FA

1 solution

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