Alg9.4.9

From Exampleproblems

Find the value of $\log_3(4)\cdot\log_4(5)\cdot\log_5(6)\cdot\log_6(7)\cdot\log_7(8)\cdot\log_8(9)\,$

Expressing the given by applying $\log_n(m)=\frac{\log(m)}{\log(n)}\,$

we get

$\frac{\log(4)}{\log(3)}\cdot\frac{\log(5)}{\log(4)}\cdot\frac{\log(6)}{\log(5)}\cdot\frac{\log(7)}{\log(6)}\cdot\frac{\log(8)}{\log(7)}\cdot\frac{\log(9)}{\log(8)}\,$

Simlifying by cancelling the same terms in the numerator and denominator, we get

$\log_3(9)\,$

Expressing 9 in terms of 3,

$\log_3(3^2)\,$

Which is equal to

$2\log_3(3)=2(1)=2\,$

Hence

$\log_3(4)\cdot\log_4(5)\cdot\log_5(6)\cdot\log_6(7)\cdot\log_7(8)\cdot\log_8(9)=2\,$

Main Page:Algebra:Logarithmic

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Alg9.4.9"

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