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Prove that log_a b = 1/(log_b a)

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steve1234 | Student, Undergraduate | Salutatorian

Posted September 3, 2013 at 5:52 PM via web

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Prove that log_a b = 1/(log_b a)

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted September 3, 2013 at 5:57 PM (Answer #1)

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The relation `log_a b = 1/(log_b a)` has to be proved.

Let `X = log_a b`

=> `b = a^X`

Take the logarithm with base b of both the sides,

`log_b b = log_b(a^X)`

Use the property `log_b b = 1` and `log a^b = b*log a`

=> `1 = X*log_b a`

=> `X = 1/(log_ b a)`

But `X = log_a b` , which gives `log_a b = 1/(log_b a)`

This proves that `log_a b = 1/(log_b a)`

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