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### High School Mathematics3.24 Change of base of a Logarithm

 log x a = log y a . log x y. Proof: Let log x a = p, Then xp = a -------I log y a = r, then yq = a -----------II also let log x y = r, then y = xr --III From I and II we get, a = xp = yq ---IV From III, we get, y q = (xr)q = xrq ---V xp =xrq bases are equal and so indices are also equal. Therefore, p = rq log x a = log y a . log x y. This result is used to change the base of the logarithm. Note: log x a = log y a . log x y. in the above equation put x = a and y = b, we get. log a a = log b a . log a b. log b a . log a b = 1. [Since log a a = 1] Example: Show that log y x . log z y . log x z = 1. Solution: logy x . log z y = log z x [Since logy a . log x y = log x a] = logz x . log x z = logx x [Since logy a . log x y = log x a] = 1 [Since log a a = 1] Directions: Prove the following statements. Also write at least ten examples of your own. 1. Show that, log 3 2 . log 4 3 . log 2 4 = 1. 2. Show that, log b a . log c b . log a c = 1. 3. Show that, log 6 5 . log 7 6 . log 8 7 . log5 8 = 1. 4. Show that, log 8 9 . log 7 8 . log 9 7 = 1.

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