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Logarithmic properties

  1.  loga (1) = 0
So when  loga (1) = x   ⇒   ax = 1   ⇒   x = 0.
 
  2. loga (a) = 1
So when  loga (a) = x   ⇒   ax = a   ⇒   x = 1.

  3. The domain of the logarithmic function is (0, ∞)
Of course, negative numbers have no logarithm, then for eaxample, if you calculate loga (–2), it produces a conflict: log(–2) = x   ⇒   ax = –2 . Because a > 0, the expression ax is always positive for all values of x and ax = –2 gives no solution.

  4. loga(x· y) = loga x + loga y
Let loga= m and loga= n, then you get am = x y an = y. Therefore

        

  5. log(x / y) = loga x – loga y
Let loga x = m and loga= n, then you get am = x and aN = y. Therefore

        

  6. log(xn) = n loga x
Of course

        loga (xn) = loga (x · x · x ··· x) = loga x + loga x + ··· + loga x = n loga x

  7. Change of base a in base b

        

Let aN = x   ⇒    N = loga x. For the logarithm with base b you get

        

 


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