Laws of logarithms

Let $ v = \log_a b$ and $ w = \log_a c$. Then we can verify the following lows of logarithms:

  $\displaystyle \log_a(bc) = \log_a b + \log_a c$   $\displaystyle \left(\mbox{ since $a^u = a^{v + w} = a^v \cdot a^w = bc$ }\right);$    
  $\displaystyle \log_a\left(\frac{ b }{c}\right) = \log_a b - \log_a c$   $\displaystyle \left(\mbox{ since $a^u = a^{v - w} = a^v \cdot a^{-w} = \dfrac{ b }{c}$ }\right);$    
  $\displaystyle \log_a b^k = k \log_a b$   $\displaystyle \left(\mbox{ since $a^u = a^{kv} = \left(a^v\right)^k = b^k$ }\right).$    

Change of base. By (2) observe that $ u = \log_b c$ implies $ c = b^u$. Taking $ \log_a$ in the both side of the equation $ c = b^u$, we have $ \log_a c = \log_a b^u = u \log_a b$. This implies that

$\displaystyle u = \log_b c = \frac{\log_a c}{\log_a b} .
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Department of Mathematics
Last modified: 2005-09-29